The Twins signed Thornburg to a minor-league deal Friday and assigned him to Triple-A St. Paul.
Thornburg will move on to the sixth organization of his career after stops with the Brewers, Red Sox, Dodgers, Reds and Braves. He was released by Atlanta in late May after he surrendered four runs on 12 hits and five walks while striking out 10 over 9.1 innings out of the big-league bullpen. The 33-year-old will serve as organizational relief depth while he's with Minnesota.