"I'm saying unequivocally: it never, never happened," he said of the allegations during a TV interview on Friday.
He asked for a search of the Senate archives for any record of a complaint Ms Reade allegedly filed at the time.
Ms Reade made a criminal complaint to police last month.
She said she was a victim of sexual assault without naming Mr Biden. The police complaint, she said, was filed "for safety reasons only" as the statute of limitations for her claim had expired.
Mr Biden is running against Republican incumbent President Donald Trump, who has been accused of sexual misconduct by some 25 women.
The Democrat put out his statement before speaking on Morning Joe, a show on US cable channel MSNBC.
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