The Northern Ireland international picked up the injury during the Gers' 3-0 win over Motherwell on Monday.
"It's not particularly good news for Kyle. He looks to have a tear in the hamstring and all indications were that was going to be the case judging by the way he pulled up," manager Ally McCoist told the club's official website.
"At this moment he is on crutches and we think probably six or seven weeks I'm afraid."
Rangers next face a trip to Arbroath in the fourth round of the Scottish Cup on Sunday.