The Braves have made an offer to the free-agent outfielder, according to a person familiar with the negotiations. The person spoke Wednesday on condition of anonymity because no agreement had been reached.
Atlanta's offer is believed to be for one year and include deferred money. Damon's agent, Scott Boras, has said he is seeking at least a two-year deal.
The 36-year-old Damon hit .286 with 24 homers for the New York Yankees last season and likely would bat leadoff for the Braves.
Atlanta added another former Yankees outfielder in December when it acquired Melky Cabrera in a five-player deal that sent right-hander Javier Vazquez to the World Series champions.
Nate McLouth is set as Atlanta's starter in center field, and is the probable leadoff hitter if the team does not add Damon or another outfielder. Matt Diaz is expected to get most of the playing time at one corner outfield spot.
Braves manager Bobby Cox said last week he envisions Cabrera playing all three outfield spots and possibly sharing a position with Diaz.
Cox said the team is prepared to give 20-year-old outfield prospect Jason Heyward a chance to win a starting job in right field. Heyward made only a brief appearance at Triple-A last season and has fewer than 200 at-bats above Class A.
"Well, there's no reason not to give him a crack at making it," Cox said. "If he is too young, that's fine, too. He can spend some time in Triple-A. But if we think he's ready and he can help us, then let's go."
Heyward (6-foot-4, 225 pounds) has the power potential the Braves need in their outfield. He was voted baseball's top prospect by Baseball America after hitting .323 with 17 homers and 63 RBIs at three minor league stops in 2009.
"If he makes the team, he's playing - lefties, righties, everybody," Cox said.
The Braves' willingness to give Heyward an opportunity could keep them from engaging in a bidding war for Damon. According to reports, Detroit and Tampa Bay also have interest.
Boras did not return a call to The Associated Press on Wednesday. Braves general manager Frank Wren had no comment on Damon.