The tight end had said that he was 95% sure that 2012 would be his final year playing in the NFL, having been drafted in 1997.
However, a team source told NFL.com: "If we offer him the money, he'll most likely be back for one more. He'll most likely be looking to miss training camp, though."
It is thought that the 37-year-old is looking for a $7m (£4.7m) salary, over $2m more than he earned last year.
Gonzalez scored eight touchdowns for the Falcons during the regular season, before securing the first post-season with of his 16-year career.