McLeish is also aware that following some recent excellent perfomances for Blues, the so-called 'bigger clubs' may be contemplating trying to lure the keeper from Manchester City when his loan deal at St Andrews expires at the end of the season.
However, McLeish revealed that he and the Blues board have not discussed trying to make Hart's move permanent when the transfer window opens in January.
"Maybe a bigger club might have their sights on him," the Blues boss has said.
"We bring players in and try and improve them and sometimes a situation like this arises.
"We've just got to keep going with Joe, and at the end of the season see what happens. He may feel he wants to come to Birmingham.
"His form has been brilliant, and let's face it, he is looking a big, powerful bloke between those sticks right now.
"He has improved since he came to us. He probably had it there anyway, but we have probably brought out the potential.
"Joe has as good a chance as anybody else to certainly be in the England squad. I know Fabio Capello and his staff like him, so he already has an edge, and his recent performances have certainly shown he can be a candidate.
"He is conscientious, he cares, and he enjoys the support he gets. A lot of players can be sensitive and see that as a criticism, but it's constructive.
"With some of the things we tweaked earlier in the season, he has improved, he has ironed out the rough edges. He is a great pro. Right now he just needs to focus on the next game, to work hard under [goalkeeping coach] Dave Watson and continue to improve."
When asked whether any discussions have taken place about making Hart's move permanent in January, McLeish added: "We need to discuss that. We've not discussed it at board leve."