Francis, a 33-year-old from North Delta, British Columbia, was 1-2 with a 5.85 ERA this year in one start and 11 relief appearances for Cincinnati, Oakland and the New York Yankees. The 10-year big league veteran will report to major league spring training.
Dirks, 28, played in 14 minor league games this year following back surgery in March. Dirks hit .276 with 24 homers and 100 RBIs in 297 games for Detroit from 2011-13.
Toronto made the moves Friday.
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