Grossman and John Beck had been battling for the starting job throughout the preseason.
Shanahan informed the quarterbacks of his decision over the weekend.
Grossman first signed with Washington last season to back up Donovan McNabb, then started the final three games of the season. The Redskins were 1-2 with Grossman at the helm. In four games overall last year, he completed 74-of-133 passes for 884 yards with seven touchdowns and four interceptions.
He signed a one-year contract this past offseason to stay with Washington.
McNabb was traded to the Minnesota Vikings after the NFL lockout ended for a sixth-round selection in 2012 and a conditional sixth-rounder in 2013, setting up the QB competition.
Grossman went 34-for-53 through the air for 407 yards, two touchdowns and one interception this preseason, while Beck completed 30-of-48 passes for 356 yards, one touchdown and two interceptions.