Chavez, 35, was the backbone of the Chicago Cubs' bullpen in the second half of the 2018 season, allowing just five runs on 26 hits in 39 innings. He walked five and struck out 42 in that stretch.
Signed by the Rangers to a $1.5 million deal last spring, Chavez had a 3.51 ERA in 56⅓ innings with Texas before being traded to the Cubs for left-handed pitching prospect Tyler Thomas on July 19.
Chavez ranked second in the majors in relief innings with 95⅓, and his 2.55 ERA was the third lowest by a reliever with at least 90 innings in the last 10 years.
New Rangers manager Chris Woodward has said he wants flexibility in constructing his pitching staff. Chavez was a starter before becoming a relief specialist.
Over 11 seasons, he has played for nine teams. His scoreless inning in relief in the National League wild-card game against the Rockies was his first postseason appearance.
ESPN's Jesse Rogers contributed to this report.