Reports suggest Hart will be dropped for Manchester City's game against Norwich on Saturday - something manager Manuel Pellegrini would neither confirm nor deny at his pre-match press conference.
"Tomorrow you will know the answer," Pellegrini said. "It is a decision I have to take every week with 22 players of the team."
Hart was replaced by Costel Pantilimon for the 2-0 midweek Capital One Cup victory over Newcastle but McLeish - who took Hart on loan during his time as Birmingham City manager - has faith in the England keeper.
"I do believe that Joe will get over this," McLeish told Weekend Warm-Up on Sky Sports News.
"We don't know what's going to happen tomorrow. Speculation is that Pellegrini might leave him out.
"Pantilimon did really well the other night, made a great point-blank save at nil-nil, took cross balls well, but I'm sure Joe will be bursting just to show everybody he can do it. Whether he gets a reprieve from Pellegrini remains to be seen."
However, McLeish thinks Roy Hodgson may be forced to rethink Hart's position as England goalkeeper should he lose his place in the City line-up.
"Certainly Roy, in a team such as England's magnitude in world football, they wouldn't go - I don't think they would go - with a goalkeeper who's been out of his club football for a long period of time."
Asked if Hart would be returning to the team for the Norwich game, Pellegrini
had told reporters at Friday's pre-match press conference: "Tomorrow you will know the answer.
"It is a decision I have to take every week with 22 players."