Avenatti made the announcement on Twitter Wednesday morning, naming his client as Julie Swetnick. In the tweet, he described her as courageous, brave and honest.
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The attorney also tweeted images of what he said was a sworn statement from his client, adding, "Under no circumstances should Brett Kavanaugh be confirmed absent a full and complete investigation."
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Two other women have come forward with allegations of sexual misconduct against Kavanaugh.
MORE: 2nd woman claims sexual misconduct by Supreme Court nominee Brett Kavanaugh
Report: 2nd woman claims sexual misconduct by Kavanaugh
California professor Christine Blasey Ford has also accused Kavanaugh of sexually assaulting her at a high school party. Deborah Ramirez has also come forward with accusations against Kavanaugh. Ramirez said in a story published Sunday by The New Yorker Magazine that Kavanaugh placed his penis in front of her and caused her to involuntarily touch it during a drunken dormitory party.
Kavanaugh has staunchly denied that he ever sexually assaulted anyone.
In a statement just released, Kavanaugh said, "This is ridiculous and from the Twilight Zone. I don't know who this is and this never happened."
His allies, including President Donald Trump, have cast doubt on the accusations.
Ford and Kavanaugh are expected to testify separately at a committee hearing on Thursday.
The Associated Press contributed to this report.