The 39-year-old Fitzpatrick played just one game this season.
Montez Sweat was set to return to practices on Wednesday after missing four games with an injury.
Washington put Montez Sweat on the NFL’s COVID-19 reserve list Wednesday after the third-year pass rusher tested positive for the coronavirus. Sweat was one of Washington’s few unvaccinated players and therefore is ruled out of the Cowboys game. Sweat could return Dec. 19 at the Philadelphia Eagles if he remains asymptomatic.