Meester sustained the injury during training camp on Sunday, but it wasn't clear he was hurt until he missed practice on Monday night.
"We're still trying to determine how long he may be out," coach Jack Del Rio said.
It's the third injury in the last four seasons for the 31-year-old offensive lineman and the second in as many training camps. Meester missed the last four games in 2005 after tearing his left biceps tendon. He missed the first four games last season after breaking a bone in his ankle during camp, but returned to make 11 starts.
His latest injury means the Jaguars could be without one of the key cogs in a line that helped the team set a franchise rushing record with 2,541 yards in 2006.
Longtime backup Dennis Norman, who filled in for Meester in '05 and '07, will move back into the starting lineup.
"That's what you have to do," Del Rio said. "Nobody in this league is going to feel sorry for you. The next guy gets an opportunity, and we expect him to step in and play well."
Copyright 2008 by The Associated Press