[Updated at 6:24 p.m. ET] Texas Gov. Rick Perry told CNN on Thursday that there was "no apology needed" from his Republican presidential campaign to fellow GOP challenger Herman Cain, who claims Perry's campaign is responsible for what Cain calls "baseless" sexual harassment allegations against him.

"Our campaign didn't have anything to do with it," said Perry, adding he'd fire any staffer if he found out they were "passing on rumors."

[Updated at 4:55 p.m. ET] GOP presidential candidate Herman Cain denounced a claim by an Oklahoma Republican political consultant that he'd seen Cain demonstrate inappropriate conduct toward a female employee during Cain's tenure as head of the National Restaurant Association.

"This is absolutely not true," Cain told Sean Hannity on the Hannity's syndicated radio program, regarding the account that Chris Wilson made on CNN and elsewhere. "They can't prove anything ... I think this Chris Wilson guy is just trying to help out the (Texas Gov. Rick) Perry campaign."

The Perry campaign has claimed it didn't know about the allegations. Wilson is currently doing polling for a political action committee supporting Perry's bid, but the group is independent of the campaign and by law cannot coordinate with it.

Allegations that Cain engaged in sexual harassment in his past surfaced Sunday.

Cain on Thursday continued to deny the allegations and said he won't be deterred by the controversy that has dominated his front-running campaign for the Republican presidential nomination.

[Initial post, 4 p.m. ET] Republican presidential contender Herman Cain has raised $1.2 million since allegations of sexual harassment in his past surfaced Sunday, his campaign said Thursday.