NY received Cleveland's first-round pick and second third-round pick in the 2019 draft along with safety Jabrill Peppers.
In August, the Giants signed him to a five-year, $90 million contract extension, which included $65 million guaranteed.
"I don't feel like I'm being given the opportunity to be the very best that I can, to bring that every single day - and that's really all I want to do, to bring that every single day, " Beckham said.
Fan and journalist Kimberley Jones said she had spoken to Odell, following his conversation with Giants general manager David Gettleman.
The deal is also second trade completed in the past week by the Giants and Browns.
In recent weeks, Beckham has been commenting and liking Tom Brady's Instagram posts, which got many in New England dreaming of him coming to Foxboro given the Patriots' needs at receiver. We can haggle about whether the compensation was enough for NY all day, but the Browns didn't destroy their season by giving up a pair of picks and a safety, they didn't mortgage a cent of their future and they landed a game-changing, potential Hall of Fame wide receiver.
The Giants own the No. 6 and No. 17 overall picks in the 2019 draft as a result of the trade. But consider the other side of that: The cap hit is less painful in a rebuilding year, and it stands to reason that the return compensation the Giants received now is better than they would have gotten last year.
He added that leaving the Giants is "bittersweet, but it is what it is".
Since joining the Giants as the 12th overall pick in the 2014 draft, Beckham has made three Pro Bowls and been one of the top receivers in the National Football League when healthy. The Giants could be considered the "new Cleveland Browns" while the Browns can proceed to build with Odell Beckham.
Dorsey has had a busy offseason, signing troubled running back Kareem Hunt to a one-year contract, trading for Vernon and agreeing to a three-year, $39 million deal with defensive tackle Sheldon Richardson.
NY could use the pick they get from the Browns to acquire Rosen, which still gives them the sixth overall pick to take another talented player.
Beckham was limited to four games in 2017 because of ankle injuries.