The 30-year-old Montero was the Rangers closer for most of last season after Jose Leclerc was sidelined for the season due to a right shoulder injury after just two appearances. Montero appeared in 17 games and was 8-for-8 in save situations. Montero was 0-1 with a 4.08 ERA.
The Rangers will receive 17-year-old Jose Corniell, a right-handed pitcher whom Seattle gave a $630,000 bonus in the last international signing period. The 6-3, 170-pound Corniell has a fastball that reaches the mid-90s and could bump even higher as his body fills out.
Montero should be a significant boost to Seattle's bullpen that was among the worst in the American League last season.
"Rafael is a solid addition to our ongoing efforts to fortify the bullpen," Seattle general manager Jerry Dipoto said in a statement. "We see his outstanding stuff and quality performance since converting to a full-time reliever as impactful in our bullpen."
Montero suffered a complete tear of the ulnar collateral ligament in his pitching elbow in March 2018 while with the New York Mets. He underwent Tommy John surgery and missed that season. In parts of four seasons with the Mets, Montero appeared in 58 games with the Mets and had 30 starts. He was 6-16 with a 5.38 ERA during his time in New York
Overall, Montero is 8-17 with 242 strikeouts, a 4.93 ERA and 8 saves over six major league seasons.
ESPN's Jeff Passan and The Associated Press contributed to this report.