Tonkin agreed to a minor-league deal with the Rangers on Tuesday, Brandon Warne of ZoneCoverage.com reports.
Tonkin had previously spent parts of five seasons with the Twins, throwing 146.1 innings of relief with a 4.43 ERA. He pitched for the Hokkaido Nippon-Ham Fighters in Japan last season, finishing with a 3.71 ERA in 51 innings.