The two sides continue negotiating a contract and have made enough progress for each side to believe the deal will get done this week, more likely sooner rather than later, league sources said.
Falcons general manager Thomas Dimitroff told the Atlanta Journal-Constitution on Monday that the talks between his team and Umenyiora were "fluid and amicable."
Umenyiora had 43 tackles and six sacks last season for the New York Giants. Following the 2010 season, in which he had 11.5 sacks and 10 forced fumbles, Umenyiora clashed with general manager Jerry Reese over his desire for a new contract.
Umenyiora, who has a home in Atlanta, discussed his talks with the Falcons with an Atlanta radio station last Friday.
"There was nothing better than being able to win there [in New York]," the free-agent pass-rusher said on Atlanta's 92.9 The Game. "The only thing I can think of, for myself personally, [better] than being able to win in New York would be if I was able to win here at home in Atlanta. So we'll see how that plays out. Hopefully, it will work out in a positive direction."
Information from ESPNNewYork.com's Ohm Youngmisuk was used in this report.