Hunter Biden, 44, had been a reserve officer starting in 2012 but failed the drug test last year, said the official familiar with the case, who spoke on condition of anonymity.
In a statement, the vice president's younger son acknowledged his discharge and expressed regret.
"It was the honor of my life to serve in the US Navy, and I deeply regret and am embarrassed that my actions led to my administrative discharge," he said in the statement, issued through his lawyer.
There was no public response from the vice president's office.
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Hunter Biden, who held the rank of ensign, had been selected for a commission in 2012 and assigned to a unit in Norfolk, Virginia to work in public affairs. He was discharged from the naval reserves in February 2014, according to a US Navy spokesman.
The details of Biden's discharge could not be released under the provisions of privacy law, Commander Ryan Perry said in an email.
The expulsion was first reported by the Wall Street Journal.
The Wall Street Journal reported that he needed a second waiver because of a drug-related charge that he received when he was younger, which officials say is not uncommon.
The news of Hunter Biden's discharge follows a series of embarrassing incidents for the US vice president, who recently had to apologize to Turkey and the United Arab Emirates after he said they had financed Sunni jihadists.
The vice president's other son, Beau, is the attorney general of Delaware, where he is standing as a candidate for governor. Beau Biden is a major in the Delaware National Guard.