The club's 2-0 defeat to Everton on Sunday - their 11th in the Premier League this season - ensured that it is now mathematically impossible to qualify for next season's Champions League.
Moyes remained defiant after United's latest setback, and insisted that he is still planning for a summer overhaul of the squad.
"We'll try and do everything we possibly can to get a team which is capable of winning better than we have done recently," he said.
However, it appears that the Glazers' patience has finally run out, with the Daily Mirror reporting that Ryan Giggs will be asked to take charge as caretaker boss for the remainder of the season.
The club have not yet confirmed Moyes' dismissal, but it is thought that he will depart before Saturday's game against Norwich.
Holland manager Louis van Gaal, who is leaving after the World Cup, Borussia Dortmund boss Jurgen Klopp, and Paris Saint-Germain's Laurent Blanc have been tipped as possible contenders to replace Moyes.