Minor, 29, is 44-42 with a 3.93 ERA over six seasons with the Atlanta Braves and Kansas City Royals. He underwent shoulder surgery in May 2015 and did not pitch in the big leagues that year or in 2016.
But he returned to log a 2.55 ERA and 1.01 WHIP and strike out 88 batters in 77⅔ innings of relief with Kansas City this season, and he was a prime target of several clubs as a free agent.
Terms of Minor's deal weren't immediately available. He joins starting pitcher Doug Fister as the second addition to Texas' pitching staff this offseason.
The Athletic was first to report Minor's pending deal with the Rangers.