Heath, 26, spent the 2011 season with Charlotte, going 4-7 with a 4.73 ERA (54 ER/102.2 IP) and 117 strikeouts in 30 games (16 starts). Heath, who averaged 10.3 strikeouts per 9.0 IP, began the season in the bullpen but made 16 of his last 18 appearances as a starter.
Following the season, the 6-foot-4, 215-pound Heath pitched for Aragua in the Venezuelan Winter League, going 2-2 with a 2.37 ERA (8 ER/30.1 IP) and 29 strikeouts in six starts.
Heath owns a 25-31 record with a 4.25 ERA (253 ER/536.0 IP) in 153 games (75 starts) over five minor-league seasons. He was signed by the White Sox as a free agent on April 22, 2010 after originally being drafted by Atlanta in the 19th round of the 2006 draft.
Leesman, 24, went 10-7 with a 4.03 ERA (68 ER/152.0 IP) and 113 strikeouts in 27 starts with Birmingham in 2011. The 6-foot-4, 210-pounder led the Barons in victories and strikeouts and was named to the Southern League midseason All-Star Team.
Leesman has gone 37-18 with a 3.63 ERA (189 ER/468.0 IP) in 85 games (83 starts) over four seasons in the White Sox minor-league system. He was selected in the 11th round of the 2008 draft.
With the moves, the White Sox 40-man roster is at 36.