But the free agent is unlikely to settle in Atlanta, ESPN's David Aldridge reports.
Dampier, a 6-foot-11 center who averaged 12.3 points and 12 rebounds last season, is weighing interest from the Hawks against a sign-and-trade offer from the New York Knicks.
According to Aldridge, it is unlikely that Dampier would accept Atlanta's offer, which is approximately $1 million less per season than what the Knicks offered to him.
The Knicks have structured a sign-and-trade deal with the Warriors that would have Golden State sign Dampier to a six-year contract, starting at $9 million per season, and then trade him to New York for forwards Nazr Mohammed and Othella Harrington.
Dampier, 29, opted out of the last two years of his contract to become a free agent, leaving more than $16 million on the table. He is coming off the most productive season in his eight-year career -- averaging a double-double while also blocking 1.85 shots in 32.5 minutes per game.
Dampier ranked fourth in the league in rebounding and seventh with 42 double-doubles. He has hired a new agent, Dan Fegan, who is based in Los Angeles.
Information from The Associated Press was used in this report.