Bryant, 30, suffered the injury late last week while lifting weights, a source told ESPN's Adam Caplan. The expected recovery time after the surgery is five to six months.
"First off, thank you to all the fans, friends, and family who have reached out expressing their well wishes in response to the news of my injury," Bryant said in a statement released by the team. "While I'm frustrated and disappointed about having to miss some time out there with my teammates, I take comfort in knowing that preparations have already begun for an epic return."
An eight-year veteran, Bryant led the Browns with a career-high six sacks last season and has started all 41 games he has appeared in over three seasons with Cleveland.
Bryant recorded 33 tackles and a forced fumble in 14 games last season, missing Weeks 3 and 4 with a shoulder injury.
"We feel awful for Des," Browns coach Hue Jackson said. "This is a young man that worked extremely hard this offseason to prepare himself for the 2016 season. He really bought in to our coaching, our team-over-self philosophy, and was intent on becoming a leader for our team.
"Ray [Horton] and the defensive staff were expecting a really productive season from him. We all understand that injuries are part of this profession, and as a team you have to find a way to overcome them. There is no doubt that we will miss Des this season, and we all wish him a speedy recovery."
The Browns, who ranked fifth-worst in the NFL with 29 sacks in 2015, will already be without pass-rusher Armonty Bryant to start the season. Bryant was suspended four games for violating the NFL's policy on performance-enhancing substances.