Clevinger suffered a partial tear of the medial meniscus in his left knee while training this week and has been on crutches since Thursday, sources told The Athletic's Zack Meisel.
The right-hander is expected to return to major-league activity in six-to-eight weeks, Meisel adds.
Cleveland will establish a timetable for the pitcher's return following the procedure.
The 29-year-old hurler has pitched well for the Indians during the last three seasons, amassing a 37-16 record with a 2.88 ERA and 3.27 FIP in 440 innings as a starter.
Clevinger is expected to miss the start of the season, according to USA TODAY's Bob Nightengale.