Clevenger fell to the ground in pain when he struck out to end Saturday's loss to San Francisco. He injured the oblique muscle.
The Cubs said he will be placed on the 15-day disabled list Tuesday when second baseman Darwin Barney is expected to return from a deep cut on his left knee.
"As soon as I did it, I knew what it was," said Clevenger, who missed a month last season because of a right oblique strain. "It was a lot worse than last year, pain-wise. Last year, I did it in the last swing of (batting practice). I went up into the clubhouse and I was a little tight. But (Saturday), I just felt like someone stuck a knife in my side and knocked the breath out of me."
Clevenger has appeared in eight games and has one hit in eight at-bats.
The Cubs agreed Sunday to a minor league deal with reliever Kevin Gregg. The right-hander had 23 saves with the Cubs in 2009 and was replaced as the closer by Carlos Marmol.
Gregg was 3-2 with a 4.95 ERA for Baltimore last season. He was with the Los Angeles Dodgers in spring training and was assigned to their minor league camp late last month.