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The news outlet reported that Musk arrived in Palo Alto, California, in 1995 to attend Stanford University but never enrolled in his graduate studies programme there.
Instead, he developed software company Zip2, which sold in 1999 for around US$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 to maintain a valid work authorisation as a student.
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 effort in US history if he is reelected.