"It's time for you, Herman, to quit," said Ginger White at a press conference in Manhattan. "You are a liar and you don't deserve the public's confidence in such an important position."
White's lawyer, Gloria Allred -- who also represents a second woman, Sharon Bialek, who first claimed in 2011 that Cain had sexually harassed her -- said her clients are prepared to testify publicly against Cain in the Senate if he is formally nominated.
The 2011 allegations, which Cain continues to deny, effectively ended the businessman's Republican presidential campaign that year.
Allred told reporters that White would "identify certain parts of Mr. Cain's body" to corroborate her account.
In her remarks, White added: "I ask Herman Cain, if I never had a sexual relationship with you, how would I be in a position to describe parts of your body that are not visible?"
Cain did not respond to a request for comment from CNN.
He said in a Fox News appearance after White's press conference that her claim was "unfounded."
"I'm not going to drop out because they are recycling the unfounded accusation from nearly eight years ago," Cain told Fox News host Neil Cavuto on Thursday afternoon. "I did know her. I don't deny that. I deny an affair."
President Donald Trump said earlier this month that he intends to nominate Cain, who previously served on the board of the Fed's Kansas City regional bank, to a seat on the powerful Board of Governors, which sets interest rates.
But four Republican senators have said they would vote against Cain if he's nominated, in part because of the allegations from White and Bialek, leaving him no clear path to confirmation.
White House officials have said Cain remains in the vetting process and has the President's support.
Trump has also said he intends to nominate his former campaign adviser Stephen Moore.
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