Callahan added that the failed physical is a result of scalp surgery that Williams underwent in the offseason, per Rhiannon Walker of The Athletic.
The Redskins now have two weeks to find a helmet that doesn't hurt the offensive lineman, reported NFL Network's Ian Rapoport.
Rapoport added that if the team can't find a helmet, Williams can either play in pain or the team can place him on the non-football injury list.
The news comes a day after Williams ended his holdout. Despite his return to the team, the seven-time Pro Bowler reportedly doesn't plan to play this season.
Washington was unable to find a trade partner prior to Tuesday's deadline but will shop Williams again during the offseason, according to Rapoport.
The 31-year-old Williams was drafted fourth overall in 2010 and hasn't played in 2019. He started 13 games for Washington last season.