The deal is worth $10 million, a source told ESPN's Jeff Passan, confirming a report by The Athletic.
Grossman, 31 spent the past two seasons with the Oakland Athletics. He hit .241 with eight home runs and 23 RBIs in 51 games in the shortened 2020 season.
"I've been fortunate to play for some great franchises during my career, and I couldn't be happier to add the Tigers to that list," Grossman said. "I've been in Detroit a number of times as a visiting player, and know how passionate the city and its fans are for Tigers baseball. I'm very excited to get to work with my coaches and teammates, and help bring winning baseball back to the Motor City."
Prior to his stint with the A's, Grossman played for Minnesota and Houston. His .826 OPS last season was only two points shy of his career high in 2016. He also set a career high with 11 homers in that 2016 season with the Twins, although he'd have had a good chance to break that if last year had been a full season.
"Robbie is a proven major league hitter and will bring consistent, competitive at-bats to our lineup," Detroit general manager Al Avila said. "His name quickly came up when our front office began planning potential offseason additions, and all of us are looking forward to having his veteran presence both on the field and in the clubhouse."
The Tigers have largely stuck to smaller deals in free agency during their rebuild. Detroit hasn't made the postseason since 2014.
In a corresponding move, the Tigers designated outfielder Troy Stokes Jr. for assignment.
The Associated Press contributed to this report.