Clippard said the injury was diagnosed as a "capsule sprain" and his shoulder will be reevaluated at the end of six weeks of rest. The right-hander signed a $2.25 million, one-year deal with the team in February.
Clippard was injured while pitching against the Milwaukee Brewers on Friday in Phoenix. He gave up seven earned runs without retiring a batter and was struggling with his command and velocity before eventually motioning for manager Torey Lovullo to take him out of the game.
Clippard, 36, has been remarkably durable throughout his career and has thrown in 777 career games, which ranks second among active MLB pitchers. He had a 2.77 ERA in 26 appearances with the Minnesota Twins last season.
"I've never been on the injured list in my career," Clippard said on Sunday. "I pride myself on being that guy who is healthy and now I'm not, so I'm bummed."