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Mr. Cain also joked that it could be smart for the other candidates to hire women to accuse them of sexual harassment as a way to boost fundraising. He also said his wife told him that Ms. Bialek’s description “doesn’t even sound like you, and I’ve known you for 45 years.”

“My own wife said I wouldn’t do anything as silly as what that woman is talking about,” he said.

The Cain camp continued Tuesday to question Ms. Bialek’s character  — first in an email blast and then at the news conference in Scottsdale, Ariz., highlighting what they called her “long and troubled history” in the courts, her difficulty in maintaining a job and problems with her personal finances.

Asked by a reporter whether he would take a lie-detector test, Mr. Cain answered, “Yes, I absolutely would,” and added, “I am not going to do that unless I have a good reason to do that.”

Asked whether he believes sexual harassment is real, he said it is a “very serious charge” and relayed that he has “seen situations where women have attempted to sexually harass men.”

Polls suggest that the allegations have not dented his support among Republican voters, though some influential conservatives are beginning to raise questions.

On Monday, the president of Concerned Women for America said the accusation by Ms. Bialek “gives me pause.”

Mitt Romney, a fellow Republican presidential candidate who is neck-and-neck with Mr. Cain in the latest USA Today/Gallup poll, said Tuesday that the allegations are going to “have to be addressed seriously.”

“I don’t want to suppose truth or lack of truth. I just think it’s important to recognize that a number of women have come forward with concerns; this woman’s charges are particularly disturbing and they’re serious,” the former Massachusetts governor said in an interview with ABC News and

Without providing solid evidence, Mr. Cain has responded with accusations of his own. He has made the case that Rick Perry’s presidential team was responsible for leaking the allegations to the media, said that the “smear campaign” against him is partially driven by race and suggested Tuesday that “the Democratic machine” pushed Ms. Bialek to make her accusations.