O'Neill said: "Birmingham have made an inquiry and they have also made a bid. The bid as it stands hasn't been accepted.
"There have also been other enquiries which haven't been followed up with a bid.
"I would love to keep Craig but he wants to play and if the two clubs can agree on things, he probably would go."
Gardner is understood to be keen to leave Villa Park and make a cross-city switch to St Andrew's in pursuit of regular first-team football.
The 23-year-old has featured only three times under Martin O'Neill this season - starting the 1-0 Europa League qualifying defeat at Rapid Vienna in August and the 1-0 Carling Cup third-round home win over Cardiff in September, while he also played as a late substitute in the 5-1 demolition of Bolton in early November.
Wolves and Stoke are also believed to be interested in Gardner.