He suffered the injury early in the second quarter of Monday night's 29-7 loss to Indianapolis.
He returned briefly, but later was replaced by reserve Quinn Gray.
Coach Jack Del Rio said the damage was discovered during an MRI Tuesday.
"David will be out this week, and potentially, he could be out as much as a month," Del Rio told reporters at a news conference.
"We will rally 'round Gray and do our best to compete as a team."
Gray will be the starter Sunday when the Jaguars travel to Tampa Bay, and the team did not say if any of the three was signed.