Cleveland is the second team expressing interest in Berry. The five-time Pro Bowler previously visited the Dallas Cowboys.
The Kansas City Chiefs released Berry on March 13 after nine seasons. The 30-year-old has appeared in just three regular-season games over the last two years due to heel injuries.
The Browns have also been in touch with safety Tre Boston's agents, Dorsey said. Boston spent last season with the Arizona Cardinals playing under then-head coach Steve Wilks, who's now the Browns' defensive coordinator.