Although there has been no official confirmation, it is understood Neville was at the renamed Aon training complex on Thursday morning for the first full day of training since Moyes was installed as Sir Alex Ferguson's replacement.
It would represent yet another switch from Everton following the earlier appointments of Steve Round, Chris Woods and Jimmy Lumsden.
However, in Neville's case it also brings a vast amount of experience of life at United.
Neville made 386 appearances in 11 seasons as a first-team player with the Red Devils.
During that time he won six Premier League titles, three FA Cups and the Champions League before leaving to join Everton in 2005, where he overcame some initial scepticism to become a crowd favourite.
Neville confirmed his intention to retire at the end of last season, although new Everton boss Roberto Martinez had claimed he intended to speak with the 36-year-old about a coaching role.
Source: PA
Source: PA