The 22-year-old England international keeper is currently on a season long loan deal at St Andrews and Blues have been told by City that they have no intention of selling Hart.
However, McLeish is keen on trying to tie up a deal to keep Hart on loan for another season.
"We want to keep him, we'd like to see him stay for another season. He wants to stay," the Blues boss has said.
"City are his club, in terms of the ownership of the contract, and we respect that.
"But we hope at some stage that City can give us a bit of goodwill.
"They have a top class keeper in Shay Given and, unless someone comes in with a ridiculous offer for Given, you can see him staying at the helm with City for the next two or three years.
"It would have to a loan deal because City have said under no circumstances will they sell Joe."