Grimm's deal includes an invitation to training camp, where the three-time defending American League Central champions will try to patch together a bullpen missing key components from past seasons.
The 30-year-old Grimm has gone 20-23 with a 4.98 ERA in 306 major league games with the Rangers, Cubs, Royals and Mariners. Grimm was on Chicago's 2016 postseason roster and appeared in three games against the Indians in the World Series.
He recorded at least 50 relief appearances with the Cubs over four straight seasons, pitching in a career-high 73 games in 2014.
Grimm split last season between the Kansas City and Seattle organizations. He twice went on the disabled list with the Royals.
The Indians lost reliever Andrew Miller as a free agent in the offseason. Closer Cody Allen remains unsigned, and the team's career saves leader is unlikely to return.