The person spoke Sunday to The Associated Press on condition of anonymity because the deal hadn't been announced.
Another person familiar with the situation said Saturday night that the Tigers and Damon agreed on an $8 million, one-year deal, subject to a physical.
The 36-year-old Damon would bring another left-handed bat to the Tigers' lineup besides switch-hitter Carlos Guillen. Guillen and Damon would likely alternate between left field and designated hitter, but manager Jim Leyland says Guillen would be his everyday hitter out of the No. 5 slot.
Leyland said he told Guillen last fall that he would be the full-time left fielder. Guillen, who batted .242. with 11 homers and 41 RBI in 81 games last season, was told he would be in the lineup every day.
Damon has also been courted by the Chicago White Sox and Atlanta Braves since leaving the world champion New York Yankees as a free agent. He batted .282 with 24 homers for the Yankees last season and is expected to lead off for the Tigers, who have a gaping hole at the top of the batting order.
"I'm happy for him that he's found out where he's going to go," Yankees manager Joe Girardi said Sunday. "I think for players that's an anxious time. Trying to figure out where you're going to be. Usually it doesn't go on this long. I wish Johnny luck. Johnny was good for us here. We all miss Johnny. This is the hard part of the game when those relationships change a little bit. I'll root for Johnny except when he's playing against us."
AP Sports Writer Larry Lage contributed to this report.