The Yankees claimed Tonkin off waivers from the Mets on Thursday.
Tonkin was designated for assignment by the Mets on Monday.
The Mets claimed Tonkin off waivers from the Twins on Wednesday, Abbey Mastracco of the New York Daily News reports.
Tonkin was designated for assignment Saturday. He gave up two runs over two innings with five strikeouts but two hits and two walks in Friday's loss to Detroit.
Tonkin was acquired by the Twins from the Mets on Tuesday in exchange for cash considerations.
Tonkin was designated for assignment by the Mets on Friday, Anthony DiComo of MLB.com reports.
Tonkin (0-1) took the loss Monday as the Mets were downed 5-0 in 10 innings by the Tigers, getting charged with five unearned runs on only two hits in an inning of relief. He didn't walk or strike out a batter.
Tonkin will begin the season on the Mets' 26-man roster, Abbey Mastracco of the New York Daily News reports.
Tonkin has a 0.00 ERA through 5.1 Grapefruit League innings with a 6:2 K:BB, but he isn't guaranteed a spot in the Mets' Opening Day bullpen, Will Sammon of The Athletic reports.
Tonkin will be on Atlanta's roster for the NLDS versus the Phillies, Mark Bowman of MLB.com reports.