The 27-year-old Olt made the Cubs' opening-day roster, but played in only six games before fracturing his right wrist. He combined to hit .273 with nine homers and 25 RBIs in 62 games with Double-A Tennessee and Triple-A Iowa.
The White Sox announced the move on Saturday.
Olt hit .160 with eight homers and 33 RBIs in 89 games last season with the Cubs. Olt was a supplemental first-round draft pick by the Texas Rangers in 2010.