Hawkins reportedly signed an offer sheet from the Browns last week worth $13.6m (£8.2m) over four years, leaving the Bengals five days to decide whether they would match the deal and retain the wide receiver.
However, Bengals head coach Marvin Lewis has chosen to allow Hawkins to leave the team, paying tribute to the 28-year-old while also stressing that his side have the depth to deal with his absence.
"I don't want to let Andrew go without making note that he has been a very hard working and contributing player. He has set a great example of being a great teammate," Lewis is quoted as saying by ESPN.
"But we're fortunate to have a lot of depth at wide receiver, and we're excited to move forward. Our leading receivers from last year will all return, and we have another group who know our system and have earned our confidence that they are ready to see more action.
"It's a given that we will invest in players to the limit of the salary cap, but the system means teams have to make choices, and we don't believe this match, at this time, would serve our goal of allocating resources roster-wide for the best possible team come September."
Hawkins was ruled out for a large chunk of the 2013 season, playing in just eight games and recording 12 receptions for 199 yards and no touchdowns.