Rosen, the No. 10 pick in the 2018 draft, last played for the Atlanta Falcons in 2021, completing 2 of 11 passes with two interceptions.
The Browns, who face the possibility that starter Deshaun Watson could be suspended under the NFL’s personal conduct policy, are looking to enter training camp with four quarterbacks.
Rosen, 25, has spent time with five NFL teams since he was drafted out of UCLA. Rosen will likely battle Josh Dobbs to by Brissett’s backup.