Citing "multiple people familiar with the situation," the newspaper said Shanahan would be dismissed Sunday or Monday after the Redskins wrap up their third losing season in his four years at the helm.
Should he be let go, the team would begin a search for its eighth head coach since owner Daniel Snyder purchased the franchise from the Jack Kent Cooke estate in 1999.
The Redskins, who entered Sunday's final with a 3-12 record, had no official comment on Shanahan's status, the Post said.
The firing would come after reported frayed relations between the coach and quarterback Robert Griffin III. Shanahan was cited in a ESPN report this month as being displeased about the QB's close relationship with Snyder.
Shanahan, after being hired in 2010 with great expectations, guided Washington to the 2012 NFC title, but has compiled an 24-39 record overall.