Reed went 5-4 with a 3.79 ERA in 71.1 innings pitched last season. His 40 saves ranked fifth in the American League.
Davidson, 22, batted .280, with 17 home runs and 74 RBI in 115 games with Class AAA Reno in 2013.
“As a young, power-hitting third baseman, Matt has put up impressive offensive numbers at every stop in the minor leagues, often as one of the youngest players at his level,” said Rick Hahn, White Sox senior vice president/general manager. “We view him as potentially a middle of the order, run-producing third baseman for a long time.”Davidson appeared in 31 games with the Diamondbacks last season, hitting .237 with three home runs and 12 RBI.