The Redskins didn't release terms, but a league source told NFL Network's Jason La Canfora that Grossman's deal is for one year.
Grossman spent last season in Houston, where he worked under new Redskins offensive coordinator Kyle Shanahan with the Texans. Shanahan now is an assistant to his father, new Redskins coach Mike Shanahan.
"Rex has some experience with Kyle Shanahan in Houston," Redskins general manager Bruce Allen said. "He wanted to give us an opportunity, and we're glad to give him one."
Allen deflected a question about whether Grossman would be a backup in Washington or have a chance to compete for the starting job, saying: "We're going to continue to try and add competition for every position."
Before signing with the Texans, Grossman spent 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 and 33 touchdowns with 36 interceptions.
"We want competition," Mike Shanahan said. "As you look at Rex's background, you can see what he's done, the experience he has. He, obviously, has a year into the system. I'm looking forward to seeing all these quarterbacks -- what they can do on the field."
The Associated Press contributed to this report.