Griffin was injured late in last week's game against the Baltimore Ravens, suffering what was diagnosed as a sprain of a knee ligament. Kirk Cousins, like Griffin a rookie, filled in well after Griffin's injury.
Griffin this week said he was able to move well and plant his right leg to make passes. Shanahan said his quarterback "looked more comfortable" in Thursday's workouts.
"I don't know yet," Shanahan said when asked who would start at quarterback. "We'll get a chance to see how Robert progresses during the week and we'll make a decision probably on game day."
Griffin has started all of Washington's games this season. The Redskins have won their last four games in improving to 7-6, one game behind National Football Conference East-leading New York Giants.
Washington plays at 5-8 Cleveland Sunday.