The England Under-21 international has joined City for the 2009-10 campaign from Manchester City.
Hart is desperate for a chance to play regular top-flight football and enhance his chances of earning a place in the senior England squad.
But McLeish has told Hart he will have to fight it out for a spot with last season's first-choice goalkeeper Maik Taylor.
McLeish told Birmingham's official website: "Joe comes in and it will be a fantastic competition between him and Maik, two great goalkeepers.
"He obviously hasn't come in to sit on the bench but I am not making any guarantees, it's as simple as that.
"We have to see how the lads do and how the competition affects them. I'm sure the two of them will handle it, being part of the goalkeepers' union.
"Joe lost his place to Shay Given last season and probably felt a little hard done by, but he got on with it, didn't hold any grudges with Shay. In fact I think the two of them are great pals.
"It's just one of these things that happen in football and you have to deal with it mentally, as well as the physical side of things."
McLeish wants his third goalkeeper, Colin Doyle, to go out on loan next season to gain more first-team experience.
The 24-year-old was kept out of the side last term due to the form of Taylor but he still played a bit part in the promotion success towards the end of the campaign.
He came off the bench to deputise for Taylor after the Northern Ireland international's red card against Plymouth and then kept a clean sheet in a vital win at Watford when his team-mate was suspended.
McLeish said: "We feel that 'Doylie' needs more first-team experience and if we can get him out on loan, that's the game plan for Colin.
"He came in and did a very good job for us last season in Maik's absence. I thought he displayed extreme professionalism to come in out of the cold and he showed what he was capable of.
"But he is still a bit raw in terms of first-team experience at the top level, which is why we've brought Joe Hart in to be the big competition there."