Jenas made the temporary switch from Tottenham to Villa Park on transfer deadline day in August.
But the former England international was initially laid low by a thigh injury and then an Achilles problem.
Jenas has been given the green light to step up his rehabilitation work after seeing a specialist on Monday but is still some way from being available for selection.
McLeish said: "I certainly know from the club doctor that he's not as bad as we first feared.
"But the only disadvantage Jermaine has is that he's short of match fitness which he won't get until he starts playing games.
"I would have thought he's three or four games away yet. He's gutted about it."
McLeish is optimistic defender James Collins will be available for Saturday's home derby clash with West Brom.
The Wales international missed the 4-1 defeat at Manchester City at the weekend with a groin problem.
McLeish said: "He has done some individual work this week and we believe he will be ready to join in with the group on Thursday."
Source: Team Talk