The right-hander suffered the injury Thursday while training at the team's spring facility in Goodyear, Arizona.
The team has not announced a time frame for his return.
Clevinger avoided arbitration this offseason after reaching a $4.1 million deal with the Indians. The right-hander went 13-4 with a 2.71 ERA in 21 starts and is being counted on to be a key part of the rotation after the Indians traded two-time Cy Young Award winner Corey Kluber.
The Indians on Friday also signed free agent Domingo Santana. The outfielder, who made $19.5 million last season, had a promising start to 2019 before injuries started to limit his playing time. He appeared in 121 games and hit .253 with 21 homers and 69 RBIs. But following the All-Star break, Santana appeared in just 31 games and hit .128.