Dann needed a fitness test before playing in Tuesday night's Carling Cup semi-final clash at West Ham. But the former Coventry player had to be substituted at half-time of Birmingham's 2-1 first leg setback.
Now he looks certain to miss at least Sunday's derby with Aston Villa at St Andrew's.
McLeish said: "We hope Scott isn't too bad but he has definitely torn a hamstring. He passed a fitness test on the morning of the cup game.
"Sometimes players want to play through even a wee niggle. Sometimes they get away with it but sometimes they can end up with a bad injury.''
Dann, who was playing non-league football four years ago, has formed an effective partnership with Roger Johnson since making the move from Highfield Road during the summer of 2009.