A league source familiar with the talks said a three-year deal was being discussed between the two sides, as first reported by the National Football Post.
Babineaux, who turns 33 in October, has had an extremely productive career with the Falcons since being drafted in the second round out of Iowa. He finished 2013 with 49 tackles, a forced fumble and two fumble recoveries. He leads all defensive tackles since 2005 with 49.5 tackles for loss.
The Falcons entered Tuesday with the mindset of landing two external free-agent linemen: guard Jon Asamoah and nose tackle Paul Soliai. While Asamoah appears to be a done deal, Soliai is sure to have a number of suitors.
Players have been informed of the Falcons' intentions to use more of a 3-4 base defensive alignment next season, but that all depends on putting together the proper personnel. Babineaux, while probably not ideal for a 3-4 set, could thrive in any system. Not to mention he's a great asset to the team as a locker-room leader.