Meester had surgery Friday to repair a torn biceps, and Jaguars coach Jack Del Rio said he will miss at least eight weeks.
"We're talking about having a shot at him by week four, week five, certainly for our bye week, which comes in week seven," he said. "We're going to get him for the last 10 games of our regular season and hopefully we earn our way beyond that."
The 31-year-old offensive lineman sustained the injury during training camp Sunday.
It's the third injury in the last four seasons for Meester and the second in as many training camps. He missed the final four games in 2005 after tearing his left biceps tendon. Meester also missed the first four games last season after breaking a bone in his ankle during camp.
Information from The Associated Press contributed to this report.