On Thursday, the team brought in the 31-year-old defensive end for a meeting and took a physical.
Now, according to Adam Schefter of ESPN, the Falcons are looking to wrap up a deal with Umenyiora by the end of the weekend.
Falcons making strong push to sign Giants free-agent DE Osi Umenyiora and are trying to wrap up a deal with him before end of the weekend.— Adam Schefter (@AdamSchefter) March 22, 2013
Dan Benton of Giants 101 had reported that the former Giants defensive end has been looking for more than market value on a deal, which is around $5 million.
It wouldn't be the first time Umenyiora had a dispute over a contract, as the one he had with the Giants in 2010 went public, and nearly held out of training camp because of it.
In the summer of 2012, Umenyiora and the Giants worked out a restructuring deal that would pay him more in salary and make Umenyiora a free agent at season's end.
Umenyiora is in a free agent class of defensive ends like Dwight Freeney, Elvis Dumervil and John Abraham; veterans looking for one last payday in the NFL.
The Falcons decided to not bring back Abraham and would be using Umenyiora to replace him at defensive end.
In 2012, Umenyiora had six sacks and two forced fumbles with the Giants, his 10th seasons with the team in which he won two Super Bowls with the team.
Umenyiora is a former second-round pick of the Giants from back in the 2003 NFL Draft and has 75 career sacks during that time.
In a conversation on Twitter, Falcons wide receiver Roddy White asked Umenyiora when he was going to sign with the team, subtly trying to lure him to Atlanta.
@roddywhitetv we will see what happens!!!— Osi Umenyiora (@OsiUmenyiora) March 21, 2013
@osiumenyiora are you close to signing with us yet— Roddy White (@roddywhiteTV) March 21, 2013
If Umenyiora drops his asking price in regards to a deal for 2013, it'll look like he will be changing his blue and white colors for black and red.
Stay tuned, Giants fans.