The injury is not expected to require surgery, according to the report.
Tate left Sunday’s 30-27 season-opening loss to the Pittsburgh Steelers after a 25-yard run in the second quarter. He finished with 41 yards on six carries (6.8 average).
Rookies Terrance West (16 carries for 100 yards, 6.3 average) and Isaiah Crowell (five carries for 32 yards, 6.4 average, and two touchdowns) ran wild while filling in for Tate in the second half. The Browns will count on them in Tate’s absence.
Tate, 26, signed a two-year, $6.2 million contract with the Browns in March. He spent the past four seasons with the Houston Texans.
Tate suffered four broken ribs last October and played through the pain for seven games before sitting out the last two. A broken ankle forced him to miss his entire rookie season in 2010.