Dobbs served as the backup to Jacoby Brissett for the first 11 games this season but did not see the field.
He has played in six games over the last four seasons with the Pittsburgh Steelers.
During the 2022 preseason with Cleveland, Dobbs completed 35-of-53 passes, throwing for 338 yards and two touchdowns. The former college standout at Tennessee added 57 rushing yards and a touchdown on the ground.
He didn’t play during the regular season. The Browns cut Dobbs’ after Deshaun Watson returned from suspension.