The Mirror says Villa owner Randy Lerner is set to move for Moyes to avoid an embarrassing managerial U-turn.
Villa were left red-faced on Friday when Roberto Martinez opted to stay at Premier League minnows Wigan.
Now Lerner could be forced to return to Villa's original short-list of McClaren, Benitez and Mark Hughes.
However, Villa's American owner is also a long-term admirer of Moyes, who has been at Everton since 2002.
Villa know it would take around £7million in compensation to the Goodison Park club to get their man - and his large backroom staff.
US tycoon Lerner would also have to free up millions for transfers to lure the 48-year-old away from Everton, where he has a limited budget.