Moyes, who was given a six-year contract after the retirement of Sir Alex Ferguson, was handpicked by the Scot to take charge of the reigning English league champions.
Moyes's backroom staff of Steve Round, Phil Neville, Jimmy Lumsden and Chris Woods are also likely to leave.
United released a short statement Tuesday morning saying: "Manchester United has announced that David Moyes has left the club. The club would like to place on record its thanks for the hard work, honesty and integrity he brought to the role."
Moyes has had a torrid tenure at the Manchester giants. Taking all competitions into account, United have won just 27 of 51 matches, drawing nine and losing 15.
The final straw for Moyes was Sunday's 0-2 defeat against former club Everton, their 11th loss this season, which left them 13 points off the Champions League places.
Moyes's only silverware at the club came back in August when they beat Wigan 2-0 in the Community Shield.
Netherlands coach Louis van Gaal former United centre-half and Paris St-Germain coach Laurent Blanc have already been linked with the job.
Giggs has made 962 United appearances since making his debut as a 17-year-old in March 1991. He also has 64 international caps for his country.