Mughelli later sent a Twitter message, thanking his fans for their support.
The 31-year-old Mughelli was due to earn a base salary of $3 million in 2012 before Tuesday's move. His 2011 season ended when he hurt his right knee in Week 7 at Detroit forcing him to be placed on injured reserve.
Mughelli, a powerful blocker who had few opportunities to run or catch the ball in his five seasons with Atlanta, made the Pro Bowl in 2010. He was a fourth-round pick by Baltimore in 2003 and played four seasons with the Ravens.
"I said that my time in Baltimore was very special when I was a youngster. But my time here in Atlanta was twice as special," he said. "I got the chance to play with a great like Tony Gonzalez. Get coached under Mike Smith and have some special relationships ... with Michael Turner, Jason Snelling and even Jacquizz [Rodgers]. Itâs been a pleasure working with those guys and winning games."
Mughelli didn't rule out returning to the team with a restructured contract, saying "I can definitely help some team win whether if itâs the Falcons or somewhere else."
The Associated Press contributed to this report.