Mourinho, who has been available since Chelsea sacked him in December, is one of the game’s most successful coaches.
He won three league titles in two spells at Stamford Bridge, and guided Porto and Inter to Champions League victories in 2004 and 2010 respectively.
The Portuguese also led Real Madrid to the Spanish title in 2012, and his appointment would see a resumption of his rivalry with former Barcelona boss Pep Guardiola, who takes over at Manchester City this summer.
Is Mourinho the right choice?
Schmeichel thinks so but said the club will have had reservations over the 53-year-old’s appointment.
Mourinho was sacked by Chelsea in December after they made a poor start to their title defence, losing nine of their first 16 matches.
He was also involved in a high-profile dispute with club doctor Eva Carneiro, who was dismissed and pursued a constructive dismissal case against the club.
“Mourinho has fallen out with a director of football at a club, a president, a team doctor,” Schmeichel, 52, told BBC Radio 5 live.
“He has done things that are not in the way Manchester United want their football club to run.”
However, the Dane thinks Mourinho can turn United into a major force again.
“He will be prepared and ready to do it, but he has to go back to the values of the football club – a very strong team consisting of exciting players who create chances and score goals,” said Schmeichel.
“All that has been missing. Now they have a guy that is supposed to be the biggest manager in the world, the one that really could save Manchester United. It has been a brave decision but one that is long overdue.”
Schmeichel also thinks former boss Sir Alex Ferguson and club legend Sir Bobby Charlton will have been consulted and given their approval to Mourinho.
“Without them saying ‘yes’ to him, it would not happen,” said Schmeichel.
He also found the net twice in a 3-2 win over Arsenal and became the youngest scorer in a Manchester derby in the Premier League era with his winner against City in March.
England beat Turkey 2-1 on Sunday and have two more friendly matches – against Australia and Portugal on 2 June – before their first Euro 2016 Group B match against Russia on 11 June.
Hodgson added: “There is no doubt I think I made the right decision putting him in this large group because the lad has enormous talent and you will know from my past record that I’m not afraid to give young players a chance. I don’t dismiss them on the basis of their age.
“But you will also know full well that where he is playing we actually have a lot of competition for places and there isn’t an awful lot of chance for him to stamp his authority on myself and the coaching staff, because there is such a short period of time.”
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Arsenal midfielder Jack Wilshere played 66 minutes in the victory over Turkey, despite starting just once for Arsenal all season having suffered a broken leg in September.
“With Arsenal I’ve been playing out wide, which is not me,” Wilshere, 24, said.
“I’m not a wide man, but Roy seems to have faith in me in the middle, so I’m happy with that. I like to play anywhere in the middle.
Meanwhile, Leicester striker Jamie Vardy ended the season with a Premier League winners’ medal and scored 24 league goals in the club’s remarkable campaign.
“I believe that the foundations are firmly in place to enable the club to move forward and achieve even greater success,” said the Dutchman, 64.
In a statement confirming Van Gaal’s sacking at 20:30 BST on Monday, executive vice-chairman Ed Woodward praised Van Gaal’s “great professionalism and dignity” and thanked him for his “excellent work”.
The club added that a “decision on a successor as manager will be announced soon”, with Mourinho, 53, and his agent Jorge Mendes expected to be in Manchester for talks with senior United officials on Tuesday.
Assistant coaches Max Reckers, Frans Hoek and Albert Stuivenberg have also left, while the club have made assistant manager and club legend Ryan Giggs an offer to stay.
It is not yet known what role that would be and Giggs is considering the offer.
Van Gaal gave debuts to 14 academy graduates, including midfielder Jesse Lingard, who scored the winner in Saturday’s FA Cup final, and striker Marcus Rashford, who has netted eight goals since making his debut in February, earning a call-up to the England squad.
“He leaves us with a legacy of having given several young players the confidence to show their ability on the highest stage,” added Woodward.
Former Ajax, Barcelona and Bayern Munich boss Van Gaal has spent over £250m in his time at Old Trafford, achieving a fourth-place and fifth-place finish in the Premier League in his two seasons in charge. They missed out on Champions League qualification to neighbours Manchester City.
Van Gaal’s two years in stats
Manchester United scored 49 Premier League goals this season – their lowest since the 1989-90 season
For the second consecutive season, United ended up with a negative goal difference away from home
They went 11 home games without scoring a first-half goal earlier this season
They conceded the joint-fewest goals in the league – 35 – along with Tottenham
The team made 3,222 backwards passes – the most in the league
Van Gaal used 33 players over the season – the second highest number after Liverpool (34)
United topped the clean sheet table alongside Arsenal with 18
In a statement released by the club at the same time his dismissal was announced, Van Gaal added: “It has been a pleasure to work with the players and it has been particularly rewarding to see so many young players take their chance to break into the first team and excel. I look forward to watching the continued development of these young players next season.”
This season’s 49 league goals is United’s lowest total since 1989-90 and they went 11 home games without scoring a first-half goal earlier this season.
Having discussed severance terms earlier on Monday – Van Gaal is believed to earn around £6.4m a year – United officially confirmed his departure at 20:30 BST.
“It has been an honour to manage such a magnificent club as Manchester United, and in doing so, I have fulfilled a long-held ambition,” he said in a statement.
“I am immensely proud to have helped United win the FA Cup for the 12th time in the club’s history. I have been privileged during my management career to have won 20 trophies but winning the FA Cup, which is steeped in so much history, will always be one of the most special achievements of my career. “
Despite being subject to chants of “attack, attack, attack” at Old Trafford this season and being booed by supporters when lifting the FA Cup trophy at Wembley, Van Gaal added: “I would also like to express my gratitude to the amazing United supporters. They are truly the best fans in the world.”
His statement also thanked everyone at the club from directors – including former manager Sir Alex Ferguson and club legend Bobby Charlton – through to the catering team and kit and laundry department, adding “never in my 25 years as a manager have I been so well supported in my role”.
Van Gaal arrived at Manchester United’s training ground at 08:45 BST on Monday and left shortly before 17:00.
United delayed announcing the decision to allow Van Gaal to leave the country, and he has now gone to the Algarve, Portugal.
Mourinho was sacked by Chelsea last December, just seven months after leading the London club to the Premier League title.
The Portuguese agreed a deal in principle with United before the FA Cup final.
Former Porto, Inter Milan and Real Madrid boss Mourinho is one of the game’s most successful coaches, winning three league titles in his two spells at Stamford Bridge, as well as guiding Porto and Inter to Champions League victories in 2004 and 2010 respectively.
He also led Real to the Spanish title in 2012 – and his arrival will mean a mouth-watering resumption of his rivalry with former Barcelona manager Pep Guardiola. The Spaniard will take over at United’s arch-rivals Manchester City this summer.
Timeline of Van Gaal’s sacking
Saturday, 21 May. 20:00 BST – Manchester United win the FA Cup
Saturday, 21 May. 20:54 BST – BBC Sport breaks the news that Van Gaal will be sacked, with Jose Mourinho in line to replace him
Saturday, 21 May. Post-match – Van Gaal’s wife informs him of sacking reports after reading BBC Sport website
Monday, 23 May. 11:21 BST – United send their last tweet for nine hours, about Jesse Lingard
Monday, 23 May. 12:00 BST – BBC Sport breaks the news that Van Gaal has been sacked
Monday, 23 May. 20:30 BST – The Old Trafford club confirm Van Gaal’s sacking
Former Dutch player, Ronald de Boer, who played under Van Gaal at Ajax and Barcelona, told BBC Radio 5 live: “He should retire and enjoy his life. He does not have anything to prove to the football world as he has an unbelievable number of trophies and has respect from a lot of players and managers.”
Ian Sterling, chairman of Manchester United Supporters Trust, said: “Van Gaal’s fate was sealed with his results over the season. The style of play and football has been the worst I have seen in my time following United. The club has acted in the best possible way for its own interest and Van Gaal has been a victim of that.”
Former Chelsea winger Pat Nevin told BBC World Service: “Van Gaal will be upset, but when time passes and he has time to look back on his career. he walks out of Manchester United after winning a major trophy. Maybe it wasn’t the worst time to go after all.”
On Mourinho, Nevin added: “Nobody but nobody in world football brings in the trophies like Jose Mourinho.
And former Manchester United goalkeeper Peter Schmeichel said Mourinho “really could save” United, calling the decision “long overdue” after the club “moved a little bit backwards in terms of its philosophy and traditions”.
Former Chelsea boss Mourinho, 53, has been out of work since being sacked by the Blues in December.
He won three league titles in his two spells at Stamford Bridge, as well as guiding Porto and Inter Milan to Champions League victories in 2004 and 2010 respectively.
He was previously linked with the Old Trafford post when Sir Alex Ferguson retired in 2013, but the club appointed David Moyes instead.
Schmeichel said the club had “moved a little bit backwards in terms of its philosophy and traditions”.
“Mourinho is not an easy call to make for the owners and leaders because if he was the right man, he would have been picked before, that was the obvious choice,” said Schmeichel.
“He has fallen out with a director of football at a club, a president, a team doctor. He has done things that are not in the way Manchester United wants the football club to run.
“All these issues would have been brought up and I am sure it will have been run by Sir Alex Ferguson and Sir Bobby Charlton. Without them saying ‘yes’ to him, it would not happen. Therefore if he comes in he is a good choice.
“He will be prepared and ready to do it but he has to go back to the values of the football club – a very strong team consisting of exciting players which are creating chances and scoring goals. All that has been missing.
“Now they have a guy that is supposed to be the biggest manager in the world, the one that really could save Manchester United. It has been a brave decision but one that is long overdue.
“Mourinho knows why he has not been asked in the past, would you not then change yourself and accommodate what the job really is about at this football club?”