Clippard, who signed a minor league contract on Feb. 23 with an invitation to spring training, left an exhibition game early on March 9 against Colorado. He felt discomfort during the return to his throwing progression this week.
The former All-Star would get $2.5 million if he makes the 40-man roster. The AL Central champion Indians start the regular season on March 28.
The 34-year-old right-hander, who appeared in 73 games with Toronto last season, was trying to earn a place in a bullpen that lost closer Cody Allen and top reliever Andrew Miller in the offseason. Clippard was 4-3 with a 3.67 ERA last year with the Blue Jays.
Clippard has a 52-46 record with a 3.16 ERA since he started with the New York Yankees in 2007. He has pitched for eight teams in his 12-year career.