They are understood to have had one offer rejected for the England Under-21 player in the region of £9million.
Villa boss Martin O'Neill would only say: "I will tell you that we are very keen to try to strengthen our side between now and transfer deadline day.
"I am sure I could get one or two in anyway but it is just adding the sort of quality that I would want - and that is exactly what I will be trying to do.
"Am I a fan of Milner? He was nearly here two years ago. He nearly signed two years ago but it didn't materialise. But I never think anything is done and dusted until anything happens.
"So many things have happened to me as a player and as a manager that I would never take anything for granted. It was disappointing at the time but it was two years ago."