According to widespread reports released at the same time on Monday on websites of numerous British newspapers, Moyes is set to be fired by United and could be forced out before the end of the season.
The club hadn't made an official comment on the speculation by late Monday, but leading British bookmakers all slashed their odds on Moyes being removed.
"They're going to have to clarify this position quickly," said former United defender Gary Neville, who described the club's handling of the situation as "repulsive."
"The average manager gets sacked every 12 months and I've always felt Manchester United should be different, hold themselves up as a club that basically stands against what's happening in the game."
United is seventh in the English Premier League in defence of its title, with Sunday's 2-0 loss at Everton ensuring the club will miss out on Champions League qualification for the first time in 19 years.
Moyes was hand-picked by Alex Ferguson, who ended his 27-year reign as manager at the end of last season, and given a six-year deal at Old Trafford last May in the hope he could create another dynasty at a club that preaches the values of stability.
He guided United to the quarterfinals of the Champions League, losing to Bayern Munich, but the team's domestic form has been woeful by its high standards.
Having won the league title by 11 points last season, United is 23 points behind current leader Liverpool, 13 adrift of the Champions League places, and has slumped to a string of humiliating losses including to fierce rivals Liverpool and Manchester City at home.
United has been defeated a club record 11 times in the league and 10 times in 22 matches in all competitions in 2014, and is on course for its lowest league finish since 1990.
The deposed champion appears ready to spend considerable sums over the summer to strengthen its squad but American owners, the Glazer family, may not be willing to trust Moyes to oversee such heavy spending.
He was indecisive in the transfer window in the last off-season, bringing in only Marouane Fellaini from former club Everton for 27.5 million pounds ($43 million), and the Belgium midfielder has been a disappointment. United broke its club record transfer fee to sign Juan Mata for 37.1 million pounds ($61.2 million) in January but Moyes has been unable to regularly play the Spain playmaker in his best position.
British newspapers are reporting that Ryan Giggs, a club great who is on the coaching staff, could be Moyes' temporary replacement while Dutch coach Louis van Gaal and Borussia Dortmund coach Juergen Klopp have been linked with any possible vacancy.
With fan backing appearing to ebb away, Moyes said after the loss to Everton that United was "on track" in its transition period post-Ferguson.
"We are rebuilding," he said, "we have got things we want to do.
"The supporters have been incredibly behind the team and supported the team throughout. They realize it has been a difficult season ... They understand it has not been good, I recognize it has not been good, it needs to be better."