Juventus midfielder Paul Pogba says he will meet with club officials to discuss the possibility of a new contract.
Fiorentina winger Juan Cuadrado has revealed that he is set for talks with the Serie A club over a new contract.
The move came on Wednesday morning - little more than an hour after West Brom boss Pepe Mel's odds had been slashed in the same market.
However, as rumours began to circulate about Moyes' future, his odds to be the next top-flight boss out were slashed from 11/1 to 5/6 following support.
Nevertheless, Press Association Sport understands the Scot is still very much in charge and that United are baffled by the reports.
The club have always made it clear that Moyes will be given time to prove that he is the right man to maintain the success delivered by predecessor Sir Alex Ferguson.
Sky Bet's head of football trading Paul Wiggins said: "The money on Moyes took us by surprise this morning as the focus was largely on Pepe Mel up until about 10.30am.
"But no sooner had the West Brom boss been made odds-on, the money on Moyes became relentless, and he went in from 11/1 to 5/6 in just over an hour."
After what has been a miserable start to life at Old Trafford, the heat intensified on Moyes following Tuesday night's 2-0 Champions League defeat to Olympiacos.
It leaves United facing an uphill battle to reach the quarter-finals - the second leg takes place at Old Trafford in three weeks' time. They are 7/4 to make it through.
The reigning Premier League champions are also struggling domestically and sit just sixth in the table, 15 points behind leaders Chelsea and 11 behind fourth-placed Liverpool with a return to the Champions League looking highly unlikely. Sky Bet offer 8/1 about them finishing fourth or higher.
Former Everton manager Moyes replaced Sir Alex Ferguson in the hot seat in the summer, signing a six-year contract. While many predicted Ferguson's act would be a tough one to follow, few expected United to perform as badly as they have done.
Moyes has spent more than £60million on bringing in Marouane Fellaini and Juan Mata but many pundits feel more sweeping changes are needed to the playing staff to improve United's fortunes.
Former captain Roy Keane was among them, saying: "I think United need six top quality players. I would think David Moyes has been shocked. I think when he went into United in the summer he probably looked at the players and was expecting great things.
"Privately, he's probably been shocked at the lack of quality he's working with."
Nemanja Vidic has already said he will leave the club in the summer, while another key member of the club's recent success, Rio Ferdinand, appears to be nearing the end of his career.
Even Robin van Persie is now 30, but Wayne Rooney last week signed a new contract with the club, one worth a reported £300,000 a week.
Mel had earlier gone odds-on following newspaper reports of unrest among the playing staff at The Hawthorns.
The former Real Betis coach was only appointed on January 9 but has failed to lead the Baggies to a victory during his seven games in charge and they remain in the thick of the relegation battle, sitting 17th, just one point outside the relegation zone.