Moyes paid the price for a dreadful campaign when he was relieved of his duties at a meeting with the United vice-chairman Ed Woodward at the club's training ground yesterday morning.
Ryan Giggs, United's most-decorated player and a player/coach under Moyes, will take charge "until a permanent appointment can be made", the club said.
The 62-year-old Van Gaal, head coach of the Netherlands, is out of contract after the World Cup, and is held in high regard by senior figures at Old Trafford.
Sources close to the club have made it clear that Van Gaal's presence in Brazil this summer would not be an obstacle to the Dutchman taking over. Talks could take place over the coming days as United want to install Moyes' successor quickly.
Van Gaal has an impeccable pedigree having lifted the Champions League with Ajax and having won league titles with Barcelona and Bayern Munich. He is used to working with big-name players and attracting major signings.
Van Gaal has made it clear he wants to move to the Barclays Premier League when his contract expires and is the bookmakers' overwhelming favourite.
The Dutchman was linked with the Tottenham Hotspur job, but United's illustrious history, and a £150m transfer kitty, make the vacancy at Old Trafford far more attractive.
Juergen Klopp, the Borussia Dortmund coach, ruled himself out. "Man United is a great club and I feel very familiar with their wonderful fans," he said. "But my commitment to Dortmund and the people is not breakable."
Pep Guardiola was considered when Sir Alex Ferguson retired last summer, but the Spaniard went to Bayern Munich, where he intends to remain. "I want to stay at Bayern for two more years," Guardiola told a press conference. "Bayern are my team, I'm comfortable here."
Other potential candidates include Lauren Blanc, the Paris Saint-Germain manager who played for United for two years, Ferguson's former assistant Carlos Queiroz, whose contract as Iran coach expires in the summer, and the highly rated Atletico Madrid manager Diego Simeone.
For now, Giggs is the man charged with restoring some pride to United. He is expected to be assisted by Nicky Butt, who is currently in charge of the under-19s and also coaches the reserves. Phil Neville and Chris Woods remain in their posts as first-team coaches, but Steve Round and Jimmy Lumsden, both brought in by Moyes last summer, have been fired.