The Twins released Thornburg on Sunday.
Thornburg was outrighted to Triple-A St. Paul on Monday, Aaron Gleeman of The Athletic reports.
Thornburg was designated for assignment by the Twins on Friday.
Thornburg's contract was selected from Triple-A St. Paul on Sunday, Megan Ryan of the Minneapolis Star Tribune reports.
The Twins signed Thornburg to a minor-league deal Friday and assigned him to Triple-A St. Paul.
Atlanta released Thornburg on Sunday.
Thornburg was designated for assignment by Atlanta on Monday.
Thornberg walked two and struck out two in a scoreless eighth inning during Sunday's loss to the Marlins.
Thornburg (elbow) tossed 1.2 perfect innings Monday in his relief appearance during Monday's Grapefruit League game against the Blue Jays.
Thornburg (elbow) signed a one-year, $900,000 contract with Atlanta on Wednesday.