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.
Villa were so hopeful after getting permission to speak to Martinez that they scrapped proposed talks with rival candidates Steve McClaren and Rafa Benitez.
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.