The club announced the signing of the deal that Bradley's agent, Mitchell Butler, said on July 2 had been agreed to with the team that took him with the 19th pick of the 2010 draft.
The Celtics provided no details of the contract, reportedly worth $32 million.
With Rajon Rondo missing significant time last season while recovering from a torn knee ligament, Bradley averaged a career-high 14.9 points in 60 games. He also is an outstanding defender.
The Celtics had extended a qualifying offer to Bradley, making him a restricted free agent and allowing them to match any offer he received from another team.