The news outlet reported that Musk arrived in Palo Alto, California, in 1995 to attend Stanford University but never enrolled in his graduate studies program there.
Instead, he developed software company Zip2, which sold in 1999 for around $300 million, according to the outlet.
Two immigration law experts quoted by the Post said Musk would have needed to be enrolled in a full course of study in order to maintain a valid work authorisation as a student.
Musk did not respond to requests for comment sent to four of his companies - SpaceX, Tesla, the social media company X and The Boring Company - nor did Musk's lawyer Alex Spiro.
Musk in a 2020 podcast cited by the Post said: "I was legally there, but I was meant to be doing student work. I was allowed to do work sort of supporting whatever."
The Washington Post cited two former Musk colleagues who recalled Musk receiving his US work authorisation in or around 1997.
Musk has endorsed Republican presidential candidate Donald Trump in the November 5 US election.
Trump has for years portrayed migrants as invaders and criminals, and during his 2017-2021 presidency took stringent steps to curb legal and illegal migration.
He is promising the biggest deportation effort in US history if he is reelected.
