The Manchester City star spent last season on loan at St Andrew's, where his superb performances were key to the Blues achieving a ninth-placed finish in the Premier League and earned him a place in Fabio Capello's World Cup squad.
Hart was an unused member of the 23-man group but has now established himself as England's top goalkeeper, but McLeish revealed there was work to be done when he first arrived at the West Midlands club.
McLeish told Press Association Sport: "I was the worst critic of my goalkeepers when I played for 18 years at Aberdeen and Scotland.
"I used to get frustrated with the `keepers. We looked at Joe Hart's game last year after the first few games. I spoke to (Birmingham goalkeeping coach) Dave Watson and said `there are a few things we've simply got to work on'.
"At the time, one of the things was that Joe was punching everything. I said `Joe, you are 6ft 4in tall, you're the only guy who can use your hands in the box'.
"Dave, myself and Joe saw some video stuff and we thought it was a wee habit he had just got into. We don't know where he acquired it but it was in his game and maybe that is why Mark Hughes decided to go for Shay Given (at Manchester City)."