Shanahan said that Griffin will rest before seeing an independent neurologist Monday evening. Griffin has not experienced any dizziness or headaches on Monday, and Shanahan said that the star rookie is "feeling good."
"We should find out in the next few days exactly what happens," Shanahan said Monday. "Right now it looks good. I'm not really sure if it stays that way. The professionals will monitor his situation and let us know if able he's able to play or not. We surely have nothing to do with it."
Griffin might participate in a cardio workout Tuesday and will practice Wednesday if he continues to avoid concussion-related symptoms. Shanahan acknowledged that the No. 2 overall draft pick will play against the Minnesota Vikings in Week 6 if he is able to practice this week.
Shanahan's update came hours after backup quarterback Kirk Cousins said Griffin was "in good spirits" on Monday morning during a mandatory meeting for rookies. Cousins also said Griffin "seems to be doing well."
Griffin is not allowed to speak to reporters until he is symptom-free. He was knocked out of Sunday's 24-17 loss in the third quarter on a legal hit to the helmet by linebacker Sean Weatherspoon while scrambling near the sideline.
"In my experience, when the quarterback gets that first hit like he received, they slide a little bit sooner in plays to come," Shanahan said. "They kind of protect themselves a little bit more."
Team spokesman Tony Wyllie said Griffin was examined by a neurologist in the locker room before being sent home from the team's stadium.
About three hours after the game, Griffin tweeted: "I'm ok and I think after all the testing I will play next week."
NFL rules require a player who has been diagnosed by the team medical staff as having a concussion to sit out the rest of the game. He also can't return to practice or game action until a team doctor and independent neurologist clear him.
The NFL will review how the Redskins handled the release of information surrounding Griffin's concussion, league spokesman Greg Aiello told NFL.com. NFL rules mandate timely and accurate reporting by teams when they release injury information, and the Redskins' initial description that Griffin was "shaken up" might not meet those standards.
Shanahan addressed the Redskins' handling of the information Monday, claiming that Griffin initially knew the score and quarter of the game after being hit. Several minutes later, Griffin no longer knew, Shanahan said.
Cousins finished the game in Griffin's place, throwing a touchdown pass and two interceptions.
If there are any setbacks, Shanahan will have to choose between Cousins and veteran Rex Grossman. Grossman has been inactive all year as the third-string quarterback, a humbling experience for a player who once led the Chicago Bears to the Super Bowl.
"Kirk has been No. 2 for a reason, because we feel like he's earned that right," Shanahan said. "We feel very good about him and where he's at, but we also have a lot of confidence in Rex as well."
Information from The Associated Press was used in this report.