Bradley started the Rays' 6-2 victory over the Cleveland Guardians on Sunday and got a no-decision after pitching five innings, allowing three hits and two runs and striking out seven batters while walking five.
Bradley, 22, owns a 5-7 record with a 5.42 ERA in 17 starts with the Rays. He went 2-5 with a 6.45 ERA in 10 starts with the Bulls this season.
The Rays are hopeful that Adam will not miss more than 15 days with a left oblique strain.
Adam told the Tampa Bay Times on Saturday that Rays head athletic trainer Joe Benge estimated the recovery time "should be fairly quick."
"It'll take, obviously, at least the length of the IL stint, but my hope is not much longer than that," Adam said. "I don't know. Hopefully the shorter the better. But I feel very confident that I'll be back for the end of the season and for the playoffs."
Adam, 32, has a 4-2 record with 12 saves and a 2.67 ERA in 55 relief appearances this season.
Reuters contributed to this report.