Jenas ruptured his Achilles tendon against Manchester United in his first start for the club following his loan move from Tottenham and is expected to be out for six months, ending his season.
The 28-year-old was hoping his move to the Midlands would reignite his England career after falling out of favour at Tottenham under Harry Redknapp.
"This is tragic for Jermaine and, of course, we are all deeply disappointed for him," McLeish told the club's official website.
"To lose a player of his talent and experience is bad enough but never to have seen him play enough in the Villa shirt for our fans to appreciate the player he is, that's terrible bad luck.
"Obviously, we wish Jermaine as speedy a recovery as possible and a full recovery, too.
"He has been a great professional to have at the club since he arrived on loan in August."