The AL Central leaders made the moves before Friday night's game against the New York Yankees.
Bruney is out with inflammation in his left hip. The 30-year-old has pitched in only one game for Chicago this season, getting the win with one scoreless inning against Milwaukee last Sunday after getting called up from the minors earlier in the day.
The 27-year-old Omogrosso has never pitched in the majors. He is 0-1 with two saves and a 4.09 ERA with Charlotte this year, and has spent seven seasons in the White Sox organization.