Reuters was not immediately able to verify the substance of the woman's allegation.
Swetnick's allegations come a day before the first woman to come forward and accuse Kavanaugh of sexual assault, Palo Alto University professor Christine Blasey Ford, is set to testify before the Senate Judiciary Committee.
"Listen to the people who've known me best through my whole life, the women who have known me since high school, the 65 who overnight signed a letter from high school saying I always treated them with dignity and respect", he added.
Kavanaugh has been accused of sexual misconduct by two other women, putting his nomination for the high court at risk.
Swetnick said in the affidavit that in 1982, she was the victim of one of the gang rapes at a party where Kavanaugh and Judge was present. She also alleged that Kavanaugh's friend, Mark Judge, was also in the room during the alleged assault.
"During the incident I was incapacitated without my consent and unable to fight off the boys raping me", she said.
Avenatti has also called for an FBI investigation into Swetnick's allegations.
In a prepared statement issued earlier for tomorrow, Kavanaugh says allegations against him are "grotesque and obvious character assassinations".
Asked about the identity of the witnesses, Avenatti said "there's a number of men and women that can attest to the facts and circumstances in this declaration, and we're going to provide those witnesses to the Federal Bureau of Investigation as soon as we're contacted".
Swetnick said, in the document given to the Senate committee, she "witnessed efforts by Mark Judge, Brett Kavanaugh and others to "target" particular girls so they could be taken advantage of; it was usually a girl that was especially vulnerable because she was alone at the party or shy".
In an interview with MS-NBC Thursday morning, Swetnick said Kavanaugh should not be confirmed as the next Supreme Court justice.
"It was only within the last couple hours that they finally got back to us and said that they want to meet with my client", Avenatti said. The lawyer rose to national attention by representing Stormy Daniels in her legal battle with President Donald Trump over an alleged affair and payments to keep it quiet. Kavanaugh has a calendar from 1982 that his lawyers say helps absolve him. Her allegations are mainly about what she claims she saw Kavanaugh participating in, and how he treated other women.
But Republican Senator Lindsey Graham tweeted: "I have a hard time believing any person would continue to go to - according to the affidavit - ten parties over a two-year period where women were routinely gang raped and not report it".