Brooke Spectorsky, president and general manager of WKYC, said Kosar could have a "presence with our broadcast team," possibly during the pregame show.
Last summer, Kosar drew criticism for in-game comments about former St. Louis third-string quarterback Kellen Clemens, the Rams' receivers and assistant coach Ray Sherman.
St. Louis coach Jeff Fisher said of Kosar: "I lost a lot of respect for him. It's highly unlikely he knew anything about our football team, but felt the need to make those comments. I don't think they were justified."
The Browns apologized to the Rams and reprimanded Kosar.
Last October, Kosar was arrested in Solon, Ohio, on drunken driving charges. The police report said he refused to take a field sobriety test. Kosar told an officer he couldn't do so because he has had several operations on his knees and ankles.
According to the report, Kosar gave the officer two credit cards when asked to show his driver's license. Kosar had slurred speech and difficulty speaking.