McKenzie dominated in winning his past two starts, allowing one run and three hits with 19 strikeouts in 15 innings.
Indians president of baseball operations Chris Antonetti said the move is a precaution and he expects McKenzie's stay on the IL to be short.
"He feels good overall," Antonetti said. "He feels really good coming into the ballpark today, but we thought it made sense for a lot of reasons to give him some time."
McKenzie retired the first 23 hitters against Detroit on Aug. 15 before allowing a two-out single in the eighth inning to Harold Castro. He held the Los Angeles Angels to two hits in seven innings and struck out Shohei Ohtani three times Saturday.
The 24-year-old rookie has shuffled between the majors and Triple-A Columbus this season. He is 3-5 with a 4.83 ERA in 19 games with the Indians.
They have brought McKenzie along slowly since drafting him in 2015. He didn't pitch in 2019 because of shoulder issues and made his debut in the majors last season.
Cleveland recalled left-hander Logan Allen from Columbus in a corresponding roster move. McKenzie's next turn in the rotation was scheduled for Friday against Boston.