The person spoke Wednesday on condition of anonymity because no agreement has 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.
Braves manager Bobby Cox says the team is prepared to give 20-year-old outfield prospect Jason Heyward a chance to win a starting job.
The 36-year-old Damon hit .286 with 24 homers for the New York Yankees last season and likely would hit leadoff for the Braves.