The Belgium international seems intent on running down his contract at Old Trafford, which is up at the end of the season.
Manchester United officials fear the 29-year-old has reached an agreement in principle with Turkish club Besiktas, but Moyes is ready to throw his hat into the ring for a player he had under him at both Everton and the Theatre of Dreams.
And sources insist the Red Devils may prefer to do business with the new West Ham manager, with a view of trying to recoup a fee of around £8.5m, rather than letting him quit the club for free at the end of the season.
United have offered Fellaini a new three-year contract worth £130,000 per week, with boss Jose Mourinho also having made a personal appeal to make him reconsider his current stance.
The Manchester giants are planning one last attempt to break the impasse, although sources close to the player have indicated it will take a dramatic u-turn for Fellaini to change his mind at this late stage in the proceedings.
The Hammers have told Moyes there are funds to spend if the player comes with the requisite amount of Premier League experience, which Fellaini most certainly does, coupled with his close relationship to the Scot.