The striker underwent surgery on ruptured ankle ligaments at the start of March and the midlands club have struggled in his absence, lying just three points above the Premier League relegation zone with two games to play.
McLeish would love to have Villa's £18 million record signing back for the games against Tottenham and Norwich, but he is determined not to rush the 28-year-old's progress. However, the Scot has given new England manager Roy Hodgson a boost by adding Bent has a chance of being fit in time for the first European Championship group game against France on June 11.
"I don't believe there is a chance of him playing again for Villa this season," McLeish told the club's official website. "The doc thinks he's got a chance for the Euros, though, he's talking about four or five weeks [until he's ready to play].
"He's running and he's working individually with the physios. I'm not sure if he's at the level of striking a ball yet, but he can't be far away from that now.
"The physios will still be working on him to get him back to his peak fitness again. We have to make sure he gets back good and proper."