The 30-year-old was placed on the 15-day disabled list on April 28, and the club confirmed on Thursday night that a tear had been found. Floyd was scheduled to receive a third opinion from Altchek on Monday.
"We'll wait until the last doctor so I can think about it, pray about it and take it from there," Floyd said on Thursday. "You just look at everything and make the best decision you can.
"There's options of surgery. There's options of rehab. And everything like that. We just got to weigh the options and see what happens. Go Monday and see what happens."
Floyd was 0-4 with a 5.18 ERA in five starts this season. He left his last outing, on April 27 against the Rays, after 2 2/3 innings.
"It's tough," left-hander Chris Sale said of Floyd's injury on Monday. "We just have to be there for him and keep his head up. He's an outstanding pitcher, and an even better person."
Floyd, who is in the final year of his contract, has won at least 10 games in each of the past five seasons for the White Sox.
Andrew Simon is a reporter for MLB.com. Follow him on Twitter @HitTheCutoff. This story was not subject to the approval of Major League Baseball or its clubs.