Reed made 77 appearances for the Red Sox and New York Mets in 2017, going 2-3 with 19 saves, 15 holds and a 2.84 ERA.
What Reed lacks in electric stuff, he makes up for in durability. Over the past five seasons, the right-hander has made 342 appearances, fifth in the majors behind Bryan Shaw, Tony Watson, Cody Allen and Tyler Clippard.
Reed, 29, also is a consistent strike-thrower, walking only 2.25 batters per nine innings since 2013. He has been a setup man and a closer in seven seasons with the Chicago White Sox, Arizona Diamondbacks, Mets and Red Sox. He wasn't eligible for a qualifying offer because he was traded midseason.
The Athletic first reported the Twins' agreement with Reed, which is pending completion of a physical exam.
ESPN's Scott Lauber contributed to this report.