Moyes was dismissed by United in a face-to-face meeting with executive vice chairman Ed Woodward at 8am on Tuesday morning, but United will not name a full-time successor until the close season.
"Following the departure of David Moyes as manager, Manchester United has announced that Ryan Giggs, the club’s most decorated player, will assume responsibility for the first team until a permanent appointment can be made," United said in a statement.
Giggs made his United debut in 1991 and is likely to retire in the summer
Borussia Dortmund coach Jürgen Klopp has already ruled himself out of the running to succeed Moyes.
"Man United is a great club and I feel very familiar with their wonderful fans," Klopp said on Tuesday. "But my commitment to Borussia Dortmund and the people is not breakable."