Floyd had given up two runs in the first inning on Ben Zobrist's two-run homer. He summoned the training staff to the mound with two outs in the third inning and was removed by manager Robin Ventura. Hector Santiago came on in relief.
Floyd, 30, was placed on the disabled list twice in 2012 with elbow soreness. The second trip was for a right flexor muscle strain.
"He has a strain right now. At least that is what we have it as, a muscle toward his elbow," Ventura said. "It is looking more like a DL stint. It is kind of like he had last year. The hits just keep coming. We just have to find a way to keep going."
Floyd dropped to 0-4 with a 5.18 ERA after the White Sox's 10-4 loss.
Ventura said Santiago likely would replace Floyd in the rotation if he goes on the DL. Santiago allowed four runs on 10 hits in 3 2/3 innings of relief against Tampa Bay.
Rays catcher Jose Molina also left, exiting the game in the fourth inning after getting hit in the right knee with a pitch from Santiago.
He is day to day with a right quad contusion.
Molina was checked at home plate by Rays manager Joe Maddon and head trainer Ron Porterfield and went to first base. Before the next pitch, Molina was replaced by backup Jose Lobaton.
Molina was 0 for 1 and entered Saturday's game hitting .234.
Information from The Associated Press was used in this report.