Bradley's new deal is for $32 million over four years. The Boston Herald earlier reported the agreement.
The Celtics had originally extended a $3.6 million qualifying offer Monday that gave them an opportunity to match any outside offer that Bradley received when free agency opened.
Boston made Bradley a priority when free agency opened at midnight July 1, and the two sides, who couldn't come to an agreement on an extension before an October deadline to extend rookie pacts, moved quickly to put a deal in place.
Speaking at his Boston-based basketball camp on Monday, Bradley said: "I want to be here. And I let [the Celtics] know that. When the time comes, we'll just see what happens, and see if we come to an agreement."
Bradley is coming off a solid 2013-14 season in which he averaged a career-high 14.9 points to go along with 3.8 rebounds, 1.4 assists and 1.1 steals over 30.9 minutes per game in 60 appearances.
Bradley rediscovered the consistency in his offensive game that had disappeared after double shoulder surgery two summers ago and was a much better rebounder. His defense slipped a bit this season, reflected in him landing only a handful of votes for the NBA's All-Defensive teams despite being a second-teamer after the 2012-13 campaign.
Before Boston's final game of the season, Celtics president of basketball operations Danny Ainge said: "Avery had a good season. I've seen improvement in Avery this year. The biggest issue with Avery has just been health. He plays hard and he's had some injuries, but none of them are injuries that should prevent him from being a great player."
Bradley missed 22 games this season because of ankle woes, and he missed 31 games the prior season after the double shoulder surgery. He also missed time at the start of his rookie campaign while recovering from surgery to repair a chipped bone in his ankle.
The Celtics believe Bradley, set to enter his fifth NBA season, is still developing. He supplanted Ray Allen as the team's starting shooting guard at the end of the 2011-12 season and the team likes the way he complements point guard Rajon Rondo in the backcourt.
The Celtics drafted Oklahoma State point guard Marcus Smart with the sixth pick in last week's draft, but Ainge stressed that he envisions a potential three-guard rotation in which Bradley, Smart and Rondo log the lion's share of minutes.