United boss Sir Alex Ferguson announced his retirement on Wednesday morning and Toffees boss Moyes was immediately tipped to succeed him at Old Trafford.
Some bookies have him as short as 10/1 ON to move to the Red Devils, bringing the curtain down on an eventful 11-year stay at Goodison Park.
Fergie is a known admirer of fellow Scot Moyes, with many seeing him as being cut from the same cloth as the veteran boss.
It would still represent something of a gamble by United, given he has never won a major trophy nor managed in the Champions League.
Everton are remaining tight lipped on the situation despite the speculation whirling all around them, with Moyes himself heading down to London to see Chelsea take on Tottenham at Stamford Bridge.
Should Moyes leave – and some reports suggest an announcement within 24 hours – then Wigan boss Roberto Martinez is tipped to replace him.
The 39-year-old Spaniard, a former Wigan player, re-joined the club in 2009 after reviving Swansea City’s flagging fortunes.
He successfully kept the Latics in the Premier League in his first three seasons at the club, despite operating on a limited budget.
They are staring down the barrel this campaign though, with Tuesday’s 3-2 defeat to Swansea leaving them three points adrift of safety.
On Saturday he will lead the team at Wembley for the club’s first ever FA Cup final after a memorable run that took in a 3-0 sixth round victory against Everton at Goodison.
Martinez infamously turned down the advances of Liverpool last season and Aston Villa the year before, citing loyalty to Wigan after they stuck with him when times were tough.
The Lancashire club have previously indicated, though, they would not stand in his way should he decide to call time on his career at the DW Stadium.
If the bookies are right, and they often are, then Martinez may well be making the short trip from Wigan to Everton in the summer.
Other contenders for the job include Porto boss Vitor Pereira, Celtic manager Neil Lennon and former Chelsea boss Roberto Di Matteo.
What are your thoughts on the managerial situation? If Moyes does leave, who should replace him? Have your say below.