Multiple sources said Lee could miss four to six weeks with one source saying it could be closer to six weeks, which means he could return for the final two games of the season on Dec. 23 against the Tampa Bay Buccaneers.
On Monday, coach Jason Garrett did not have a specific timeline for his absence, but said, "we anticipate him to be out for a little bit. No question about that."
Lee pulled up in the second half of the Titans' loss and did not return. Lee missed three games earlier this season with a hamstring injury after getting hurt Sept. 23 against Seattle Seahawks.
Last year, Lee missed five games with a hamstring injury.
Rookie Leighton Vander Esch will replace Lee in the starting lineup. Vander Esch leads the Cowboys in tackles with 74, according to the coaches' breakdown. Lee has been credited with 34 tackles this season.
Injuries have been an issue for Lee, 32, since the Cowboys took him in the second round of the 2010 draft. In addition to hamstring strains, he has missed time with wrist, toe, neck and knee injuries and one game with a concussion. He has not played 16 regular-season games in his career, but the coaches kept him out of a meaningless finale in 2016.
The Dallas Morning News was first to report the likely timeline for Lee's absence.