Deborah Ramirez, a Yale university classmate of the judge, detailed the alleged incident to the New Yorker.
She alleges Mr. Kavanaugh exposed his genitals at a dormitory party and thrust them in her face, causing physical contact without consent.
The judge has denied both allegations, labelling the latest “a smear”.
Professor Christine Blasey Ford, who last week came forward as the woman alleging Mr. Kavanaugh sexually assaulted her in 1982, has agreed to testify about the allegation at a Senate committee hearing on Thursday.
The choice of a new justice for the Supreme Court is pivotal, as it often gives the final word on highly contentious laws, with an immense impact on US public life.
Ms. Ramirez alleges the incident occurred when she was a freshman at Yale University alongside Mr. Kavanaugh during the 1983-4 academic year.
She says they were both taking part in a drinking game at a dormitory party where people sat in a circle and selected others to drink.
She says at one point, a plastic penis was pointed in her direction by one man, and later another exposed himself directly.
“I remember a penis being in front of my face,” she said. “I knew that’s not what I wanted, even in that state of mind.”
Ms. Ramirez said she was mocked and taunted when she remarked it wasn’t real, and then ended up touching the genitals while attempting to push the man away.
“I wasn’t going to touch a penis until I was married,” she said, referring to her devout Catholic upbringing. “I was embarrassed and ashamed and humiliated.”
She says she remembers Mr. Kavanaugh standing to her right and laughing, pulling up his pants.
The article says the 53-year-old acknowledges gaps in her memory caused by alcohol that night, which is said to have made her hesitant to come forward when contacted.
The journalists behind the piece, say they contacted dozens of classmates to try and corroborate Ms. Ramirez’s account.
Three former students, who were not independent eyewitnesses, recalled being told or discussing the incident in vague terms at the time.
Two males allegedly involved in the incident disputed her account alongside several other classmates, adding the allegation against Mr. Kavanaugh was “completely out of character”.
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In a statement, the Supreme Court nominee strongly refuted the allegation.
“This alleged event from 35 years ago did not happen. The people who knew me then know that this did not happen, and have said so,” he said.
The Aquarius: Migrant rescue ship has registration revoked
When the Aquarius next docks, it will have to remove its Panama maritime flag and cannot set sail without a new one.
It is the last private rescue ship operating in the area used for crossings from Libya to Europe.
The charities who run the vessel accuse the Italian government of pressuring Panama into deflagging the Aquarius.
The two groups who lease it, Médecins Sans Frontières (MSF) and SOS Mediterranée, say they were notified of the decision by the Panama Maritime Authority (PMA) on Saturday.
The authority is said to have described the ship as a “political problem” for the country’s government, and said Italian authorities had urged them to take “immediate action” against them, according to SOS Mediterranée.
Italy’s Interior Minister Matteo Salvini, who has previously described the aid boats as a “taxi service” for migrants, denies his country put pressure on Panama.
On Sunday, he tweeted he “didn’t even know” what prefix Panama has for telephone calls.
Mr. Salvini has been a prominent figure in a public immigration crackdown in Italy since his government, a coalition between the right-wing League party and the anti-establishment Five Star Movement, came to power in June.
