The deal will be completed when Mesa passes a physical exam this week, the sources said.
Mesa, a 40-year-old right-hander, ranks 13th on baseball's career list with 320 saves. He went 1-5 with a 3.86 ERA last year for the Colorado Rockies in his 18th major-league season.
The Tigers, who have Todd Jones as their closer and the hard-throwing Joel Zumaya and Fernando Rodney in setup roles, were looking to add a veteran arm to their bullpen. They're also interested in acquiring a left-handed reliever after losing Jamie Walker to Baltimore through free agency.
Jerry Crasnick covers Major League Baseball for ESPN Insider.