The right-hander suffered the injury Wednesday during a spring training workout at the team's facility in Goodyear, Arizona.
The team has not announced a time frame for his return.
Clevinger, 29, avoided arbitration this offseason by reaching a $4.1 million deal with Cleveland. He went 13-4 with a 2.71 ERA in 21 starts in 2019 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.
He has won 38 games over the past three seasons. He made 21 starts in 2019, but missed time early in the year with a strained back muscle.
Cleveland on Friday also signed free agent Domingo Santana to a $1.5 million contract that includes a $5 million club option for 2021. The outfielder, who made $1.95 million with the Seattle Mariners last season, had a promising start to 2019 before missing 24 games in August and September with an injured right elbow.
Santana appeared in 121 games and hit .253 with 21 homers and 69 RBIs.
Prior to signing with the Mariners, Santana spent three-plus seasons with the Milwaukee Brewers. His breakthrough season came in 2017, when he hit 30 home runs with 85 RBIs, 88 runs and 15 stolen bases for the Brewers.
Since entering the majors in 2014, Santana has a career .259 batting average with 75 home runs and 232 RBIs.
The Associated Press contributed to this report.