Crain hasn't pitched since April 20 and has made six appearances with a 2.57 ERA this season. He threw about 20 pitches on the side Thursday and said he continued to feel some discomfort in his left oblique. He and White Sox manager Robin Ventura were hopeful he wouldn't have to be put on the disabled list.
The 26-year-old Axelrod was 2-1 with a 1.08 ERA and had allowed three runs and struck out 26 batters in 25 innings during his four starts this season for the Charlotte Knights. He ranks second in the International League in ERA, third in strikeouts and fourth in innings pitched.
"He has been up here so it's not like bringing someone up that can't pitch here," Ventura said. "We have confidence in (him) so we will just make do until Jesse is ready."
Axelrod, who will be used as a long man in the bullpen, pitched four games, including three starts, for the White Sox last season. He was 1-0 with a 2.89 ERA.
ESPNChicago.com's Bruce Levine contributed to this report.