The club are now looking for their seventh full-time manager since being taken over by the Pozzo family in June 2012.
The Pozzo family also own Italian club Udinese and Spanish side Granada.
West Ham manager Slaven Bilic called the decision “sick” and “crazy”.
He added: “I’m sorry for Quique. It’s completely wrong, they’ve had a fantastic season.”
Geoff Doyle, BBC Three Counties Radio sports editor
“Expect tears at Vicarage Road on Sunday as Quique Sanchez Flores waves goodbye to Watford fans. Let’s be clear, Flores does not want to leave. It was the owners and board who made the decision.
“Supporters are divided, but the majority agree with the split. They are delighted to be playing Premier League football again next season thanks to Watford’s form up to Christmas.
“As stupid as it sounds, Flores may have done too well in the first five months of the season and expectation was cranked up. Results in the new year have been worrying and so has their defensive style of play.
“It is hard not to feel that the job he had done up to that point had been more than good enough. But football is all about the here and now. For a lot of Watford fans, the damage has been done.
Manchester United defender Matteo Darmian is expected to miss out again with an ankle injury.
Adnan Januzaj is out with a hamstring injury, Marouane Fellaini is banned, while Bastian Schweinsteiger and Luke Shaw remain out through injury.
Bournemouth have no new injury problems but manager Eddie Howe says he might rotate his squad on Sunday.
Defender Sylvain Distin and Adam Smith both remain out with back and hernia problems respectively.
MOTD COMMENTATOR’S NOTES
Guy Mowbray: “Like the whole country, Manchester United don’t know whether they’ll be in or out of Europe after the summer.
“All the options are possible, with their best-case Champions League (play-off round) scenario dependent on the double of a home win AND Swansea beating Manchester City.
“Bournemouth finally come to Old Trafford for a league game looking for a double of their own, having beaten United on the south coast in December.
“That’s undoubtedly one of the highlights of their first ever top division season, with the fact that they’ll be back again next year an achievement that has perhaps not had the recognition it deserves.
“Maybe one day Eddie Howe will be back at a big ground like this on a more fixed basis?”
Manchester United manager Louis van Gaal: “We are dependent on Manchester City, that’s the difference and that’s also frustrating when you had the chance two matches before the end to keep that advantage.
“We were 2-1 ahead with 15 minutes to play and then we gave it away [against West Ham].
“That is frustrating and now we are dependent (on what City do), but still it is possible so we will fight until the end.”
Bournemouth manager Eddie Howe: “It is nice to be involved in a game that means something.
“The last few games have been difficult for us because we have not been a team that are used to not having that competitive edge.
“In the last few seasons there has always been something on every game and we have become very accustomed to it and embraced the pressure and relished it in some ways.”
Manchester United still have an outside chance of a top-four finish. They should have no problems winning but I do not see Manchester City being beaten at Swansea with what is at stake for them.
The Cherries will not just be rolled over at Old Trafford, but I don’t see them stopping United from ending the season with a win.
He finished fifth in his first campaign in charge with 72 points.
But his side have won just five league games at home in this campaign and will end with their lowest points tally at Goodison Park since 1981.
Former Everton player and current under-21 coach David Unsworth and ex-boss Joe Royle, who also played for the Toffees for eight years, will take temporary charge for the final game of the season against Norwich City on Sunday.
The case for Benitez
Benitez, 56, started the season with Spanish giants Real Madrid but faces the prospect of the second tier of English football if he remains at Newcastle.
However, it is believed the Spaniard, who took charge of the Magpies in March, has a break clause in his contract.
Newcastle legend Alan Shearer says he would be “amazed” if Benitez remains at St James’ Park, but a group of Magpies fans have not given up hope.
They have issued a statement urging him to stay.
“Whilst we fully accept the short-term prospect of life in the Championship may not be as appealing as other offers, we can certainly promise you the potential rewards of the longer term would exceed any position that you consider,” it read.
Benitez, who led Liverpool to a Champions League win in 2005, is likely to have a number of offers should he decide to leave.
He also won Fifa’s Club World Cup and Italian Super Cup with Inter Milan and guided Chelsea to a Europa League success in 2013.
However, ex-Newcastle midfielder Jermaine Jenas says Benitez would not be a good fit for Everton because the fans would not like his defensive brand of football.
The case for De Boer
Frank de Boer, who left Ajax on Thursday, led the Dutch club to four straight league titles between 2011 and 2014 but missed out to PSV on the final day this season.
As a player, he won the Champions League and Uefa Cup with Ajax, before claiming a La Liga title with Barcelona.
Prior to taking charge of Ajax, the former Rangers defender, 45, was an assistant manager with the Dutch national team.
Michael Ball played alongside defender De Boer at Rangers in 2004 and believes the Dutchman would be “refreshing” for the Toffees.
“He likes to bring a lot of youth into his teams and, with the Everton squad being so young, it could only be a positive for him,” said Ball.
“I know he’s unproven in England, that’s the only negative. I think De Boer would be refreshing for the Premier League and refreshing for Everton.”
The case for Koeman
The Dutchman has impressed at Southampton after taking over from Tottenham-bound Mauricio Pochettino in June 2014.
The 53-year-old’s side will finish no lower than eighth in the Premier League this season but are hoping to better last season’s best-ever finish of seventh and reach the Europa League.
Koeman is well respected, having starred for the Netherlands and Barcelona as a player before embarking on a managerial career that has taken in Ajax, Benfica, PSV Eindhoven, Valencia and Feyenoord.
Southampton centre-half Virgil van Dijk is “confident” Koeman will stay.
“I think he has done an amazing job so far,” said Van Dijk. “Hopefully we can do it next year as well and then it is about him and the club and what he wants to do.”
The other contenders
Take your pick from Manuel Pellegrini, Mark Hughes, Brendan Rodgers, Moyes, Martin O’Neill and Unai Emery.
Pellegrini will leave Manchester City at the end of the season, while Hughes has impressed at Stoke City and Rodgers remains out of a job since being sacked by Liverpool in October.
Former Toffees boss Moyes, who spent 11 years in charge at Goodison, is also available after unsuccessful stints at Manchester United and Real Sociedad.
O’Neill will guide the Republic of Ireland at Euro 2016 but has yet to sign a new contract for the 2018 World Cup.
And Emery will secure a hat-trick of Europa League titles with Sevilla if the Spanish side beat Liverpool in this season’s final.
West Ham and Liverpool could face a play-off to determine who qualifies for the Europa League.
The Hammers lie sixth in the Premier League before the final round of matches, while Liverpool are eighth.
The Reds also play Sevilla in the Europa League final knowing that a win would secure a Champions League place.
If they have to qualify for Europe through the league, there is a scenario that has both clubs level on points, goal difference and goals scored.
How it could happen
West Ham would have to lose 1-0 at Stoke City AND Liverpool beat West Brom 2-1; or the Hammers lose 2-1 while it finished 3-2 to the away side at The Hawthorns, or so on. In addition, seventh-placed Southampton would have to be denied a win against visitors Crystal Palace.
Those scorelines would leave the London and Merseyside clubs level after 38 matches of a season lasting nine months.
If they were tied, the play-off would only be needed if their finishing spot is in a Europa League position, and Sevilla were to win next week’s final in Basle.
A Premier League spokesperson confirmed an extra game could take place, although the likelihood is small.
League rules state: “If at the end of the season either the league champions or the clubs to be relegated or the question of qualification for other competitions cannot be determined because two or more clubs are equal on points, goal difference and goals scored, the clubs concerned shall play off one or more deciding league matches on neutral grounds, the format, venue and timing of which shall be determined by the board.”
Any date or ground has not been decided upon.
Europa League qualification
Manchester City are already assured of at least a Europa League place after winning the Capital One Cup.
If they finish in the top four, that place in the Europa League third qualifying round will pass to the sixth-placed team.
The same will happen if they secure a place in the Europa League group stage by finishing fifth – and realistically, goal difference means City are unlikely to finish lower than fifth.
If Liverpool win the Europa League, they will qualify for the Champions League group stage. If Crystal Palace win the FA Cup, they will enter the Europa League at the group stage. If Manchester United win, that place will be given to the seventh-placed team.
There are no longer any Fair Play places allocated to the Europa League.
Hodgson has delayed the announcement of his squad to take to France until after the final Premier League fixtures this weekend, with the Football Association citing “fitness reasons”.
He has already lost Arsenal’s Alex Oxlaide-Chamberlain and Danny Wellbeck to injury, with Manchester United defender Luke Shaw another to miss out.
Former West Ham and Tottenham striker Defoe said he joined Sunderland last January from Major League Soccer side Toronto, with an England place at the back of his mind.
Defoe, capped 55 times by England, said: “As a forward you get judged on scoring goals. The England managers at the time thought I could always come on and score. When I came back I thought ‘all I’ve got to do is go there, focus, get my goals and hopefully get in the frame and take it from there’.”
Tottenham’s Harry Kane (25) and Leicester’s Jamie Vardy (24) are the only Englishmen with more top-flight goals this season and Sunderland manager Sam Allardyce said his striker could “still do it at the highest level”.
Defoe’s last cap was in November 2013 and he has previously said he would “never be at peace” with missing out on the 2014 World Cup squad for Brazil.
“I think he has done an amazing job so far. Hopefully we can do it next year as well and then it is about him and the club and what he wants to do,” said the former Celtic player.
Martinez, 42, promised to deliver Champions League football for Everton when he replaced David Moyes in the summer of 2013 – and he finished fifth in his first campaign in charge with 72 points.
But this season his side have won just five league games at home and are guaranteed to finish with their lowest points tally at Goodison Park since 1981.
Benitez, who took charge at St James’ Park with the Magpies second from bottom in March, has a break clause in his contract meaning he could walk away from the club if he wished following their demotion to the Championship.
Benitez started the season with Spanish giants Real Madrid but faces the prospect of the second tier of English football if he remains at Newcastle next season.
The 56-year-old led Everton’s city rivals Liverpool to a Champions League win in 2005 and guided Chelsea to a Europa League success in 2013.
Benitez’s family have remained in the north west of England following his six-year spell at Anfield.
“Rafa Benitez goes to Everton, Roberto Martinez goes to Newcastle. That’s not beyond the realms of possibility,” former Leeds United and Manchester City defender Mills told BBC Radio 5 live.
“He’s got experience, I can’t see him staying at Newcastle,” said Mills, who added that the Spaniard’s Liverpool history “doesn’t matter”.
But former Tottenham and Newcastle midfielder Jermaine Jenas disagreed, arguing that Everton fans want to see attacking football.
“With Benitez, I can feel them going too defensive and they wouldn’t be happy with that,” he added.
The Newcastle Fans’ Forum issued a direct plea to Benitez on Thursday, asking him to stay on Tyneside.
In a statement, it said: “Whilst we fully accept the short-term prospect of life in the Championship may not be as appealing as other offers, we can certainly promise you the potential rewards of the longer term would exceed any position that you consider.”
The forum members urged the club to use “every possible resource” to try to retain Benitez, citing his “ability to connect with the fans”.
He led Ajax to four straight league titles between 2011 and 2014, but missed out to PSV on the final day this season. His agent Guido Albers said De Boer would be “interested”, while his brother, Ronald, said he “was the right man for the job”.
Michael Ball played alongside defender De Boer at Rangers in 2004, but the 112-cap Netherlands international spent the majority of his playing career with Ajax and Barcelona.
Ball, who made more than 100 appearances for Everton, believes the Dutchman would be “refreshing” for the Toffees.
“He’s defensive-minded, he was a defensive player and a defensive coach,” said Ball.
“He likes to bring a lot of youth into his teams and with the Everton squad being so young it could only be a positive for him.
“I know he’s unproven in England – that’s the only negative. I think De Boer would be refreshing for the Premier League and refreshing for Everton.”
The other contenders
Manuel Pellegrini will leave Manchester City at the end of the season, Mark Hughes has impressed at Stoke City and Brendan Rodgers remains out of a job since being sacked by Liverpool in October.
Rodgers’ last game in charge of the Reds was a 1-1 draw at Goodison Park – and Jenas says the Northern Irishman’s Anfield connection would not go down well with Everton fans.
He added: “I was thinking maybe Brendan Rodgers but you’ve got the Liverpool issue there as well. Maybe Tim Sherwood?”
Former Wales forward Hughes had a spell at Everton in 2000 and has spent the majority of his managerial career in the Premier League with Manchester City and Blackburn Rovers before his move to the Potters.
Asked about Hughes’ future, Stoke goalkeeper Jack Butland told BBC Radio 5 live: “We’re confident he will stay, we’ve got a good thing going on despite the tough end to the season – and the manager’s played a massive part in it.”
Ball added: “Everyone knows what you’re going to get with Hughes. Pellegrini is a great manager but similar to Roberto with his style of play.”