United confirmed in a statement on Wednesday morning that Ferguson would retire this summer after more than 26 years in charge.
Moyes has long been tipped as a possible replacement for English football's most successful manager but the Scot is now favourite to take the Old Trafford helm, with an announcement expected within the next 48 hours.
The 50-year-old's Everton contract runs out in the summer and no agreement has been reached over a new deal.
He has previously said that he wants to wait until the end of this campaign before deciding whether to extend his time at Goodison Park into a 12th year.
Moyes has brought stability and consistency on a relatively small budget to the Merseysiders, despite not winning a major trophy, and is viewed by United chiefs as a man who, like his outgoing compatriot, values youth development and longevity.
Everton have yet to comment on the speculation, while Moyes was driven to London after training for a pre-arranged meeting with chairman Bill Kenwright.
Jose Mourinho, widely expected to replace Rafa Benitez at Chelsea, and Jurgen Klopp, who has led Borussia Dortmund to the Champions League final, also head the betting with Sky Bet, who make Moyes odds-on for the post, while former striker Ole Gunnar Solskjaer is also prominent in the market.
United chief executive David Gill has outlined the qualities needed to succeed Sir Alex Ferguson as manager amid expectation Moyes will be the new man in charge.
"The qualities are the ones that have been inherent at Manchester United for many years," Gill told MUTV.
"Our two most successful eras were with managers who got involved with all aspects of the club, from the youth team to the first team, to get that degree of loyalty and understanding of the football club.
"Clearly he has to have the requisite football experience, both in terms of domestic and European experience. It is a small pool."
Former Manchester United striker Michael Owen said he would choose the Everton boss.
Owen, the former Liverpool striker who played under Ferguson for two seasons, said on Twitter: "What an act to follow. I guess only Mourinho would have the confidence to strut through the door. If it were me, I'd go for Moyes."
Former United keeper Peter Schmeichel also believes Moyes would be a good fit as a successor as he has proved he can stay at one club for a long time, spending 10 years at Everton.
"What we are looking for is, not someone to come in 10 months or three years, we want someone to come stay there and give stability," he added.
"When we talk about Moyes, he has been a decade at Everton and done a fantastic job on limited funds.
"Is he cut from the same cloth? I dont know. Don't forget there will be major changes with David Gill stepping down - the club is really changing in the summer, the more stability they can add the better.
"David Moyes, he sounds like a good solution."