The 26-year-old has only ever played for the Giants, joining in 2014 and he signed a five-year $95m (£72.5m) contract extension last year.
Beckham Jr was rated the third best wide receiver in 2018 with 77 catches for 1,052 yards and six touchdowns.
In return, the Giants get a round one and round three pick in the 2019 NFL draft and Browns safety Jabrill Peppers.
The annual player draft starts on Friday, 26 April.
League rules prohibit teams from announcing trades until 20:00 GMT on Wednesday but an official NFL Twitter posting showed a picture of Beckham Jr in a Cleveland shirt with the message: Coming soon in Cleveland.
Beckham Jr has scored 44 touchdowns in 59 NFL appearances.