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 are facing the possibility that starter Deshaun Watson could be suspended under the NFL's personal conduct policy. Watson and Jacoby Brissett, who was acquired in free agency this offseason, will essentially both be QB1 entering training camp, and the Browns are planning to start Brissett in the event that Watson is suspended.
Rosen, 25, has spent time with five NFL teams since he was drafted out of UCLA in 2018.
ESPN's Jake Trotter contributed to this story.