Alexy signed a minor-league contract with the Twins on Dec. 19.
The White Sox released Alexy on July 1.
Alexy was sent outright to Triple-A Charlotte on Friday, Scott Merkin of MLB.com reports.
Alexy (oblique) made his spring debut in Saturday's 6-4 loss to the Dodgers in Cactus League play, retiring two batters while allowing two runs on three hits and two walks.
Alexy has been delayed by a left oblique issue this spring but could appear in a game by the end of the week, James Fegan of The Athletic reports.
The White Sox claimed Alexy off waivers from the Twins on Monday, Betsy Helfand of the St. Paul Pioneer Press reports.