Dombrowski confirmed the offer in a telephone interview with The Associated Press after telling reporters in Lakeland, Fla., that the Tigers were interested in signing the outfielder.
Damon also would give Detroit a left-handed bat it needs and a veteran in the outfield, where his savvy is an asset and his arm a liability.
The Atlanta Braves also appear interested in adding Damon.
Damon helped the Yankees win the World Series for the first time since 2000, hitting .282 with 23 homers and 82 RBIs, but he was coming off a $52 million, four-year contract.
While he initially wanted a long-term deal at the same average salary, New York valued the 36-year-old at far less. The Yankees kept reducing their offer, claiming payrolls constraints.
They now have no place for Damon after acquiring Curtis Granderson from the Tigers, signing free agent outfielders Randy Winn and Marcus Thames and signing designated hitter Nick Johnson to become their No. 2 hitter.
The Chicago White Sox withdrew their offer to Damon, general manager Ken Williams said Friday, a decision first reported by ESPNChicago.com.