The England Under-21 international has had only a bit-part role to play at Villa this season.
Wolves have also enquired about Gardner but Birmingham are the only club to have so far made a definite bid.
O'Neill said: "I know Craig wants to play. In an ideal world, I would love to keep him. He is as genuine as they come.
"There are some players who have left in my time here who haven't made a most incredible contribution.
"But I would be sorry to see Craig go but accept the fact if two clubs agree a fee, that he would probably would go.
"Birmingham have made a bid and, as it stands, it hasn't been accepted.
"There have been other enquiries but, as we speak, those other clubs haven't made a bid."
Gardner could become Birmingham's second January signing, following Sporting Gijon midfielder Michel to St Andrew's.
He would link up with two former Villa team-mates, defender Liam Ridgewell and veteran striker Kevin Phillips.
Birmingham boss Alex McLeish retains an interest in signing Ryan Babel despite an initial bid - believed to be around £8million - being rejected by Liverpool earlier in the month.
Celtic have also stressed that another McLeish target, Aiden McGeady, is not for sale.
Birmingham have flown out to Malta today for four days to get in some warm-weather training after the snow and ice disrupted their recent schedule.
Aston Villa 8/13, Draw 5/2, Blackburn 11/2