“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.
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.”
Manchester United Supporters’ Trust chairman Ian Stirling described the style of football under Van Gaal as “the worst I have seen in my time”, describing it as a “game of chess”.
He said that supporters want “someone in charge who respects what the club is about” but expressed concern that Mourinho “plays a similar style to Van Gaal”.
However, former Chelsea winger and BBC pundit Pat Nevin said Mourinho was “willing to change things and adapt”. He 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”.
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
They had the highest possession average in the league at 58.47%
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 late on Monday.
“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 of his departure 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.
He was previously linked with the Old Trafford post when Sir Alex Ferguson retired in 2013, but the club appointed David Moyes instead.
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