Damon played golf Monday with White Sox catcher A.J. Pierzynski and broadcaster Ken "Hawk" Harrelson, according to major-league sources.
Pierzynski lobbied Damon to sign with the White Sox, and Damon's wife, Michelle, would prefer him to play in a more cosmopolitan city than Detroit, multiple sources say.
But whether Damon will sign with the White Sox remains in question.
The White Sox have not budged from their original offer to Damon, according to a high-ranking team official.
That offer is not thought to be competitive with the Tigers' reported proposals to Damon -- one year, $7 million or two years, $14 million.
Some with the White Sox, however, are confident that a deal with Damon could happen, and believe it might even be close.
Clearly, the White Sox are discussing Damon at length internally and remain in contact with the player's agent, Scott Boras.
Club officials are mindful that Boras might be using them in an attempt to persuade the Tigers to increase their offer. But Damon also would be a good fit for the Sox, who are looking for both a leadoff man and left-handed hitter.
Chicago's U.S. Cellular Field is far more hitter-friendly than Detroit's Comerica Park. Damon put up big numbers last season at the new Yankee Stadium. If he did the same in Chicago on a one-year deal, he could re-enter the market in a strong position next season.