Weeden, 30, was the 22nd pick in the first round of the 2012 NFL Draft and has struggled since entering the league.
Last season, he only completed 52 percent of his passes for 1,731 yards with nine touchdowns and nine interceptions while starting five contests. He was 0-5 in those games.
Weeden has two years left on his rookie contract and is scheduled to make $1,124,872 next season and $1,492,308 in 2015.
The Browns had more turnover in the their front office on Tuesday, with general manager Michael Lombardi being fired and CEO Joe Banner saying he will be leaving the team in the next two months. The team also promoted Ray Farmer to general manager.
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"Given the turmoil with the coaching staff and lack of public support from the most recent front office," Silver was told, "he's interested in a fresh start, with a clean slate and an opportunity to compete to show he has the talent to develop into a productive player in the league."