The team confirmed the signing, announcing that running back Evan Royster was waived/injured in a corresponding move.
Grossman, 35, was a first-round draft pick of the Chicago Bears in the 2003 NFL draft. He last appeared on a regular-season NFL roster in 2013 with the Washington Redskins. He last played in an NFL regular-season game during the 2011 season, when he started 13 games for the Redskins.
The Falcons obviously felt uneasy about the backup quarterback battle between T.J. Yates and Sean Renfree. Neither stood out in the first two preseason games, and Yates has struggled both in practice and live action.
"It's all for our competition," Quinn said of signing Grossman. "We're going to constantly keep looking to battle and see where we can improve, if we can. I'm actually also looking forward to seeing T.J. and Sean going into this weekend as well. ... Every chance we get to come and battle and compete, we're going to do that, at all positions."
Quinn said Grossman will start practicing with the team Wednesday but will not play in Saturday's preseason game against the Miami Dolphins.
Current Falcons offensive coordinator Kyle Shanahan worked with Grossman for four seasons with the Redskins and also with the Cleveland Browns last season before the quarterback was released in late August.
"I loved Rex because he was so tough," Shanahan said. "Rex would hang in there, get the crap knocked out of him every week. He never flinched. He'd hang in there and take all the hits.
"He didn't mind the pressure on him. He'd let it sling every time. Sometimes I'd have to tell Rex that he was letting it sling around a little too much and that you might have to take a sack here or throw it away. But the thing I loved about Rex is how much he competed."
Grossman, who has 47 career starts, brings to the Falcons an experienced veteran presence. He has 863 career completions on 1,562 attempts (55.2 percent) for 10,232 yards with 56 touchdowns and 60 interceptions. He was the starting quarterback when the Bears lost 29-17 to the Indianapolis Colts in Super Bowl XLI.
Ryan has started all 16 games in each of the last five seasons. The last time he was inactive was Dec. 13, 2009, against the rival Saints as a result of turf toe.