The midfielder has yet to make his Aston Villa debut, since joining the club on a season-long loan from Tottenham.
Jenas could have made his debut at Everton on Saturday, but suffered a thigh strain in training on Friday.
“He got a kicking strain and there was no way we were risking him,” McLeish told the Sunday Mercury. “I think he was a bit over-zealous. He really pushed himself in training.
“He may have even felt it a couple of days ago but decided to come and train such was his keenness to be involved.
“You can’t knock a player for that. I hope it doesn’t set him back too long and he’ll be back with us within a couple of weeks because we need that competition in the centre.”