According to ESPN's Adam Schefter, "Rosen would be hesitant to come out in next year's NFL draft if he knew that Cleveland was going to take him with the No. 1 overall pick."
Schefter added Rosen "has expressed concern about winding up in Cleveland and would rather be with a more stable franchise, such as the [New York] Giants."
The Browns, who are 0-14, will secure the No. 1 pick for the second straight season if they lose to the Chicago Bears or the Giants beat the Arizona Cardinals on Sunday.
As for Rosen, the junior signal-caller is expected to sit out Tuesday's Cactus Bowl against the Kansas State Wildcats.
Citing a source, Sports Illustrated's Bruce Feldman reported Saturday that UCLA's medical staff "doesn't want him to play" in the bowl game because he suffered a concussion late in the season.
Rosen wanted to suit up, according to Feldman, but he has been advised to sit "under doctor's orders."
Once UCLA's season wraps up, Schefter reported Rosen is expected to have a "serious conversation" about his future before deciding whether he wants to declare for the draft or return for his senior season.