He has been prone to the occasional error with the latest gifting Sunderland a goal at St Andrew's on Saturday when he came for a free-kick but was beaten to the ball by Michael Turner, although it did not prove costly as Birmingham held on for a 2-1 victory.
McLeish said: "Joe had a lack of action in the game and goalkeepers sometimes get a bit impetuous.
"Joe maybe came into an area where he didn't need to come (for the free-kick).
"It is just typical of a 'keeper that they feel 'I've got to do something and get involved.'
"He is disappointed. He beats himself up. He really cares but he knows these mistakes in the Premier League are highlighted.
"If he is going to go to the World Cup with England, he has got to improve on that side of things."
Hart will be ineligible to face his parent club at St Andrew's on Sunday when last season's number one Maik Taylor will return between the posts.
Blues have collected 10 points from their opening 10 games back in the top flight and McLeish believes his side deserve to have picked up more given the quality of their overall performances.
He said: "We fell a bit behind in the points-per-game ratio before Saturday and I certainly feel we should be at least three points better off.
"Our performances have merited that and the only time I have been disappointed was the performance at Burnley.
"We have some very tough games coming up (against Manchester City and Liverpool) and, if we can take some points from those, I will be very pleased with the start to this half of the season."