It is understood Liverpool will pay an initial £23m with a further £2m in conditional add-ons.
Wijnaldum will join the Reds on their pre-season tour of America, where they will play Chelsea (27 July), AC Milan (30 July) and Roma (1 August).
He began his career at Feyenoord before moving to PSV Eindhoven and has made 30 appearances for the Netherlands, helping them finish third at the 2014 World Cup.
He scored on his Premier League debut in August 2015 and found the net a further 10 times during the season, including four in a 6-2 win over Norwich, but could not prevent Newcastle being relegated to the Championship.
The 61-year-old signed an initial two-year deal after compensation was agreed with Sunderland, whom he steered to Premier League safety last season.
He succeeds Roy Hodgson, who quit after England were knocked out of Euro 2016 in the last 16 by Iceland.
Allardyce, whose first game in charge will be a friendly at Wembley on 1 September against as yet unnamed opponents, said he was “honoured”.
He added: “It is no secret that this is the role I have always wanted. For me, it is absolutely the best job in English football.
“I will do everything I can to help England do well and give our nation the success our fans deserve. Above all, we have to make the people and the whole country proud.”
Football Association chief executive Martin Glenn said: “His excellent managerial credentials, including his ability to realise the potential of players and teams, develop a strong team ethos and embrace modern methods that enhance performance, made him the outstanding choice.
“We could not help but be energised by his personal perspective on England’s future.”
A statement on the FA’s website said Allardyce’s primary target is qualification for the 2018 World Cup in Russia, but he “is also charged with helping technical director Dan Ashworth integrate and strengthen the FA’s elite performance and coaching programme across the England senior and development teams at St George’s Park”.
Allardyce’s first competitive match will be in Slovakia on 4 September as England begin their qualifying campaign for the 2018 World Cup.
He leaves Sunderland after nine months and the Black Cats are now looking for their ninth manager in eight years.
A statement from the club read: “The focus of everyone at Sunderland AFC is on moving forward quickly and decisively, with the appointment of the club’s new manager to be confirmed at the earliest opportunity.”
Allardyce, a former Bolton, Newcastle, Blackburn and West Ham boss, becomes the 14th permanent England manager.
He has never won a major trophy but did win promotion to the Premier League with Bolton and West Ham.
Allardyce has been endorsed by his fellow managers, including Manchester United boss Jose Mourinho, former England manager Sven Goran-Eriksson and ex-Spurs boss Harry Redknapp.
Mourinho said Allardyce was “more than ready” to lead the national side, while Redknapp said he would bring “a Premier League style and pace” to the national side.
Former FA director David Davies told BBC Radio 5 live: “This is a challenge for English football. This is the person the League Managers’ Association probably would have wanted. Now will the clubs actually go out of their way to help the national team because they’ve got the person they wanted – one of their own?”
Allardyce was first interviewed for the England job following Eriksson’s departure after the 2006 World Cup, but Steve McClaren was appointed. He has been vocal about his disappointment in not being selected then.
This time he was the early favourite, chosen by a three-man FA panel of Glenn, Ashworth and vice-chairman David Gill ahead of Steve Bruce, who resigned as Hull City manager on Friday.
Bournemouth’s Eddie Howe and USA manager Jurgen Klinsmann were reported as potential candidates, but it is not known how many other interviews were conducted.
Why has he got the job?
The FA panel said it wanted a strong-minded, tactically savvy manager who could build a clear team identity.
Glenn told BBC sports editor Dan Roan the new manager would need to “build resilience” in players so they are able to deal with criticism on social media and the pressures of an “intensely passionate” English media.
Neil McDonald, who has worked alongside Allardyce at three Premier League clubs, said: “He gives the players everything they need to perform to the highest level and lets them express themselves as much as they possibly can.
“He’s been in the game a long time, he’s won a lot of games and a lot of respect off everybody and it’s well deserved to be given the England job.
“He should have had one of the big four, big six jobs in the past. But the clubs he’s gone to he’s always improved them and improved the players.”
The Allardyce effect
Allardyce is known for having an immediate impact on sides when he joins them – shown below after he arrived at Blackburn Rovers (2008-09) and Sunderland (2015-16).
The graph also shows the dip in form immediately after Allardyce has left (Newcastle 2007-08) and Blackburn (2010-11).
England have had a near-perfect qualifying record for recent tournaments, but have won just one of their past seven games in the finals.
They open the Premier League season by hosting champions Leicester on Saturday, 13 August.
Bruce began his managerial career with brief spells at Sheffield United, Huddersfield Town, Wigan Athletic and Crystal Palace before a six-year stay at Birmingham City. He twice helped the Blues win promotion to the top flight before returning to Wigan in 2007.