The Atlanta Falcons will engage quarterback Matt Ryan in negotiations to extend his contract this offseason, but sources said Tuesday that persuading tight end Tony Gonzalez to return has become a higher priority at the moment.

The Falcons have been exerting tremendous pressure on Gonzalez to make one more Super Bowl run, and sources said he's wavering from his intention to retire for the first time. While he is scheduled to become an unrestricted free agent, Gonzalez said at his locker after Atlanta's season ended in the NFC Championship Game that if he were to return he would be interested in playing only for the Falcons.

While the Falcons would certainly like to have a commitment from Gonzalez before free agency begins March 12, general manager Thomas Dimitroff said no deadlines have been established.

"Simply stated, the sooner we know, the better, and yet there are no deadlines on this for Tony," Dimitroff said. "Tony needs to come to his decision and be at peace with his decision."

Gonzalez is tempted because he knows -- barring major injuries to key players -- the Falcons have the talent to go deep into the postseason, and there has been no evidence to suggest his own performance has declined.

A 13-time Pro Bowler, who will turn 38 on Feb. 27, Gonzalez holds NFL records for touchdown receptions (107) and reception yards (14,268) by a tight end. He had 93 catches for 930 yards and 8 touchdowns last season.