Villa boss Martin O'Neill revealed that Blues had made an offer for the 23-year-old midfielder but that the bid fell short of their valuation of the player.
"Birmingham have made an inquiry and have also made a bid. The bid hasn't been accepted," said O'Neill.
"There have also been other enquiries from clubs which have not 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."