Bruney, 29, was 1-0 with a 1.31 ERA in 19 outings at Charlotte. He had seven saves and 30 strikeouts. He has 249 games of major-league experience with the Diamondbacks, Yankees and Nationals. He has a career 4.52 ERA in the major leagues.
Marquez, 26, was 3-4 with a 3.97 ERA and one complete game in nine starts with Charlotte. He made one relief appearance with the White Sox last season.
McPherson had just two hits in 15 at-bats after joining the White Sox when Mark Teahen went on the disabled list earlier this month. Harrell pitched four innings Sunday, the same day he had been officially recalled.
The White Sox's 40-man roster is at 37 after the moves.
Doug Padilla covers the White Sox for ESPNChicago.com.