The Toffees captain has announced he will leave Goodison Park at the end of the season, but could quit playing altogether with the chance to follow David Moyes to Old Trafford
Neville, who turned 36 in January, is still considering his options over whether he wants to retire from playing and move into coaching, but is being lined up for a backroom position by both clubs.
Moyes, who will be confirmed as Ferguson's replacement after the 71-year-old announced his stunning retirement on Wednesday, is keen to harness Neville's experience in the dugout at the Theater of Dreams next term.
There have also been suggestions on Merseyside that Neville could be Moyes' shock successor at Goodison Park next season, but Goal understands that Toffees chairman Bill Kenwright has earmarked Wigan boss Roberto Martinez as the club's number one target.
Moyes had publicly offered Neville a coaching role at Everton after the defender announced he would leave Goodison Park in April, stating: "If he wants to be part of my staff here he is more than welcome, he really is.
"He has been a big, powerful influence here for eight years and a big help to me in my job."
And despite Neville having insisted that he wants to continue playing at the highest level, it is thought by sources close to United that the chance of returning to Old Trafford, where he made 386 appearances under Ferguson, could make him reconsider.
Neville will join the England Under-21 coaching staff ahead of this summer's European Championships in Israel, having worked with Stuart Pearce's backroom team ahead of a qualifier against Belgium in February 2012.