7:54pm: Rosenthal takes back his previous tweet, and says that the White Sox did not make Damon a second offer. Their original offer still stands, though it is not believed to be as high as Detroit's.
Rosenthal adds that Damon and his wife "would prefer him to play in a more cosmopolitan city than Detroit," for what it's worth. What's more, Damon played golf Monday with Sox catcher A.J. Pierzynski and broadcaster Hawk Harrelson.
7:39pm: The White Sox have made a second offer to free agent outfielder Johnny Damon, tweets Ken Rosenthal of FoxSports.com. The terms of the offer are unknown, but Rosenthal says a deal may be gaining momentum.
Yesterday we heard that there was only a "very small" difference of opinion between the Tigers and Damon's camp on salary, though this new offer from the ChiSox may have changed that dynamic.