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Over a steak dinner Tuesday night in northwest Washington hosted by Georgia Sen. Saxby Chambliss, GOP presidential hopeful Herman Cain told a group of Republican senators that previous sexual harassment allegations, which have ignited a firestorm of controversy around him, were unfounded.

"He said privately what he said publicly that (the accusations) are baseless, that this has more to do with termination," recounted Sen. Richard Burr, R-NC, who attended the dinner.

Chambliss, who said he has known the Georgia pizza mogul for 15 years when both men worked in Atlanta, invited Cain to Tuesday night's gathering, a trip made by a number of the GOP candidates, including Former Utah Governor John Huntsman.

The conservative southern Republican senator, in typical low-key fashion, said he was forming no judgments about the allegations that came to light through a POLITICO report, though he made clear he was "not endorsing anybody."

"You know, we in politics know you have bumps in the road, "Chambliss said, giving Cain the benefit of the doubt, "Things like this come out that it's very difficult to explain in the press matters that can't be talked about because of a sealed document."

Chambliss said he was not sure "if this thing will handicap him at this point. Nobody knows."

Burr was equally unsure. "His campaign needs to find a way to successfully bring some finality to this," adding, "Whether this is a crippling blow is more left up to their ability to end the story and for people to be comfortable with it."

The senator said he finds "no fault" with the way the controversy has been handled by the Cain campaign, saying, "I find it hard to remember what I did 12 or 15 years ago, so that doesn't surprise me that they're in search of all the details that the media might require."

Burr told Fox he thought Cain could be "one of the Republicans who could give the president a run for his money," but only "if he makes it through the issues that are currently in front of him."

The senator said he could not conceive of a situation where the National Restaurant Association, where the sexual harassment allegedly took place during Cain's tenure as the group's president, would release the complainants from their confidentiality agreements. "I don't like the precedent that would be set by that."

Sen. Tom Coburn, R-Okla., for his part, said he was more concerned about hearing how Cain would get the nation out of its economic quagmire. "I don't make any judgments on that," ‬Coburn said, referring to the harassment charges.

Coburn heartily endorsed Cain's "9-9-9" tax code reform plan, saying "it generates a tremendous amount of growth in the country," though he said such a plan would almost certainly fail in the Senate.‬

The senator, a fiscal hawk who has warned of a looming economic disaster in the U.S., said the media is focused on the wrong thing.

‪"He's the frontrunner this week," Coburn said, referring to Cain. "Ya'll will pick someone else next week. To me it's not important now. None of that's important now. Your all's focus on a presidential campaign is a disservice to this country when we have a problem that's going to sink us. You ought to be focused on that."‬

The dinner, at Bobby Van's Grill, was attended by 11 senators, including a Mitt Romney supporter, Sen. Jim Risch, R-Idaho. Also attending - Republican senators Lamar Alexander, R-Tenn., Rob Portman, R-Ohio, Ron Johnson of Wisconsin, Tennessee's Bob Corker, Kelly Ayotte of New Hampshire, Lindsey Graham of South Carolina, and former North Dakota Governor-turned Sen. John Hoeven.

"I think they were very impressed with him. They didn't know him, and they had an opportunity to engage him," Chambliss recalled. "There was a lot more talked about than the last few days."

Cain remained in town for a series of meetings with lawmakers in the House and Senate on Wednesday, including an evening meeting with Tea Party darling Sen. Jim DeMint, a conservative lawmaker some have called a GOP "kingmaker."

DeMint told Fox that the recent allegations about Cain "certainly shouldn't" knock him out of the running. "I think it's a matter of how he handles it. It seems like what we've heard from people who have worked with him for years is the complete opposite from what's been alleged. But if there are other allegations that have come to light, who knows."