Carroll's future at Anfield has been placed in doubt by the arrival of Fabio Borini from Roma last week, and new Reds boss Brendan Rodgers is also stepping up his interest in Fulham attacker Clint Dempsey.
Dempsey's signing would surely open the door for a loan deal for the England international.
West Ham and Fulham are also in the queue..
Villa new boss Paul Lambert: "There's no doubt he's a top player, and it wouldn't be one I would dismiss if it was there.
"I've always said that the group need a hand. They need players to come in and give them a hand and we're trying to get them in."