City saw a £20 million bid for Milner, 24, rejected earlier this summer, and boss Mancini has now indicated he will return with a renewed offer for a player who Villa are thought to value at £30 million.
"Milner is a good player, but today he is an Aston Villa player," Mancini told Sky Sports News. "If we have this chance [to sign Milner] I think we can take him, but we must wait now."
City goalkeeper Shay Given, a former team-mate of Milner's at Newcastle, believes the midfielder is a quality player.
"I worked with James at Newcastle and he is a very, very good player, but we have got some great players already," said Given.
Patrick Vieira has also welcomed City's pursuit of Milner: "He [Milner] is a terrific player. Of course it is not my place to say if he has to come or not but when you look at players we have around it will be good. The attraction of City is that the future is fantastic. We have a really competitive team and I think every single player will want to play for City because the future is bright for City."
Earlier on Thursday, O'Neill had explained: "I had a conversation with James before the World Cup and he intimated that he would like to go at that time.
"I haven't spoken to him since. I know there have been conversations with his agent and our chief executive. Manchester City have made an offer for the player but certainly there is a difference in valuation, but I'm sure things will get resolved.
"James is due back next week after an extended stay after the World Cup - the same as the rest of the World Cup players like Emile Heskey, Stephen Warnock and Brad Guzan. They are all due back next week and I imagine things will get resolved one way or the other in the very foreseeable future."
Milner has two years remaining on his contract but is reported to have refused to discuss an extension.