The fresh allegation comes as the Senate Judiciary Committee is preparing to hear testimony from another woman, Christine Blasey Ford, who alleges that Kavanaugh sexually assaulted her at a high school party more than 30 years ago.
Kavanaugh denied the claims in a statement to the outlet, saying the event "from 35 years ago did not happen".
White House spokeswoman Kerri Kupec called the new allegation part of a "coordinated smear campaign by the Democrats created to tear down a good man".
"As with any witness who comes before the Senate, the Senate Judiciary Committee can not hand over its constitutional duties to attorneys for outside witnesses", Mike Davis, Senator Grassley's top nominations counsel, wrote in an email exchange with Ms Ford's lawyers obtained by the Associated Press.
In a weird turn, one of those allies, the conservative legal analyst Edward Whelan, posted pictures on Twitter Thursday night of a high school classmate of Kavanaugh's - with the man's name - and suggested he was responsible for the assault.
But the lawyers say the Republican-controlled Senate panel hasn't said who will be asking questions of Ford and Kavanaugh.
Senate Judiciary Committee Chairman Chuck Grassley had requested that Ford respond to his invitation to testify before the Senate by Saturday afternoon.
At least two Democratic senators are investigating the claim, the New Yorker reported. The counsel said it was the committee's "non-negotiable" right to determine who would be allowed to testify.
The magazine said it had not found eyewitnesses to confirm that Mr Kavanaugh was at the party, though one former classmate remembered hearing about the incident from another, and independently corroborated numerous details offered by Ms Ramirez. But after consultation with a lawyer, Ramirez told the magazine she felt confident enough in her recollection that it happened. He expressed confidence that Republicans "will manage this confirmation properly with the utmost respect for all concerned" and said he expected Kavanaugh to join the high court soon.
"If the attack on Dr. Ford was as bad as she says, charges would have been immediately filed with local Law Enforcement Authorities by either her or her loving parents", Trump tweeted.
Militants attack Iran army parade killing civilians: state media
Both Daesh and an anti-government Arab group called the Ahvaz National Resistance, claimed responsibility for Saturday's attack . The U.N. ambassador, without mentioning Giuliani, told CNN , "The United States is not looking to do regime change in Iran ".
The committee asked the friend if she would agree to have a confidential phone call with minority and majority staff so she could be asked about the allegations against Kavanaugh, according to an email exchange obtained by USA TODAY.
Days later, on a Friday afternoon in her California office, Rep. Anna Eshoo sat with Ford and listened to the story, in wrenching and hard detail.
The White House is approaching Ms Ford's potential testimony with trepidation, nervous that an emotional performance might not just damage Mr Kavanaugh's chances but could further energize female voters to turn out against Republicans in November.
"Brett was laughing", she told the magazine. For Kavanaugh to be confirmed, Republicans can only afford to lose the support of two GOP members, if all Democrats oppose the nomination.
Ford's lawyers said that, in a Sunday morning call with committee staff members, they agreed to the hearing even though the committee refused to subpoena Mark Judge, a Kavanaugh friend who Ford said witnessed the attack, as well as others she said were present.
We're going to start the program today with the latest on the nomination of Supreme Court hopeful Brett Kavanaugh.
"We are aware of significant evidence of multiple house parties in the Washington, D.C., area during the 1980s" during which Kavanaugh and others "would participate in the targeting of women with alcohol/drugs in order to allow a "train" of men to subsequently gang rape them", Avenatti wrote.
Grassley's spokesman, Taylor Foy, accused Senate Democrats of withholding information on the latest allegation and of seeking a "political takedown" of Kavanaugh.
Ford is set to testify before the Judiciary Committee about her allegation on Thursday.