Grossman spent last season with the Houston Texans, working with new Redskins offensive coordinator Kyle Shanahan.
The 29-year-old Grossman played the previous six seasons with the Chicago Bears, leading them to the Super Bowl after the 2006 season.
He has played in 37 NFL games, including 31 starts. He has completed 524 of 971 passes (54 percent) for 6,197 yards, with 33 touchdowns and 36 interceptions.
The Redskins released veteran backup quarterback Todd Collins on March 4.