Ferguson announced he would be retiring earlier today after a glittering 26-year spell in charge at Old Trafford.
Moyes, out of contract with Everton at the end of the season, is expected to be handed the daunting task of replacing the 71-year-old at the end of the season.
The Scot has established Everton as a top-six team during his 11-year spell at Goodison Park, but some questions remain over his ability to manage a team at the very top.
Real Madrid boss Jose Mourinho was tipped to take over from Ferguson earlier today - speculation fuelled by comments from Manchester United chief executive David Gill which also seemed to rule Moyes out of the running.
"He (the next manager) needs the required football experience in terms of domestic and European requirements, so it is a small pool, but we'll move forward," he said.
Moyes, 50, guided the Toffees to Champions League qualification in 2004 but saw his side knocked out in the preliminary rounds, while his 15-year managerial career is yet to yield a single major trophy.
Ferguson will call time on his managerial career after the Red Devils' final fixture of the season against West Brom on May 19.
The former Aberdeen boss steps down having guided Manchester United to their 13th Premier League title and 20th league title overall - helping them surpass Liverpool's record league haul of 18 along the way - while he also won five FA Cups, four League Cups and two Champions League trophies.
Moyes refused to comment when asked about mounting speculation regarding his future this morning.
Do YOU want to write for GiveMeSport? Get started today by signing-up and submitting an article HERE: http://gms.to/writeforgms