Johnson has remained intent on leaving Cleveland since making the request in April.
The veteran back parted ways with his former agent last week to team up with the highly renowned Rosenhaus, who engineered moves to get defensive end Emmanuel Ogbah and receiver Breshad Perriman out of Cleveland this offseason.
Browns general manager John Dorsey said he was unwilling to move Johnson - who's set to be the primary backup to Nick Chubb, at least until Kareem Hunt's eight-game suspension is up - and called him "a viable member of this organization."
Johnson's demand apparently caused friction within Cleveland's locker room. Baker Mayfield took issue with the running back wanting off the team in June, saying to reporters that Johnson should commit or "get out of the way."
The young quarterback's public comments reportedly led to a group of veteran players voicing their displeasure with how he handled the situation.
Johnson rushed for just 201 yards on 40 carries last season, but he was a big part of the passing game, hauling in 47 receptions for 429 yards and three touchdowns.