They also claimed offensive lineman Matt McCants off waivers from the Oakland Raiders.
Greco suffered the injury late in the third quarter of Sunday's 27-13 loss to the New York Giants.
He started all 12 games this season, 10 at right guard and two at center. He didn’t miss an offensive snap until he left Sunday's game with 3:54 remaining in the third quarter.
Greco has appeared in 111 career games with 70 career starts. The Browns acquired him in a trade in 2011.
The Browns have lost both of their starting guards this season to mid-foot injuries.