Joe Hart has and will return to Manchester City at the end of the season anyway when his loan deal runs out.
If Birmingham want any chance to get Hart, even on loan, they shouldn't really act as a buffer if the FA allow the move. City could repay the terms of the loan to cash strapped Birmingham, then give them first refusal should City not stick with Hart over the coming months.
Contrary to what people have suggested, Hart IS wanted at City, but with the form Given has been in they knew Hart was too good to be a no.2 and kept on the bench, which is why they allowed him to go on loan. Hart is a City keeper for the future, Given won't last forever so Hart could use this period to cement his place in the side.