Outsider Republican presidential candidate Herman Cain is spending a tumultuous week in Washington courting the GOP establishment. And, at least for now, it seems that the establishment likes him — despite allegations of sexual harassment that continue to surface.

Following a series of private meetings with the Georgia businessman early in the week, Republican Members pronounced themselves impressed with Cain and unconcerned about the sexual harassment claims that have dogged the White House hopeful since Politico reported them Sunday. Cain spoke today with Members in several private settings as he sought to boost support for his campaign, sell his controversial 9-9-9 tax overhaul and simply introduce himself to an audience of strangers.

However, a media horde trailed Cain at every stop and sought to get him to comment on sexual harassment allegations and a sealed settlement that occurred while Cain was head of the National Restaurant Association during the 1990s. While Cain was making the rounds on Capitol Hill, the Associated Press broke news that a third woman now claims Cain acted inappropriately toward her when she worked at the association, and a GOP pollster who supports Texas Gov. Rick Perry’s presidential campaign told an Oklahoma radio station that he personally witnessed Cain sexually harassing a female employee of the restaurant group in the late 1990s.

But Sen. Richard Burr (N.C.), who was among 10 Republican Senators to attend a private Tuesday evening dinner with Cain, said the presidential candidate is positioned to be a major factor in the primary if his campaign is not derailed by the harassment charges and his poor handling of the flap.

“I think he’s a very impressive candidate,” Burr said. “He’s got an impressive business record, and I think we will all wait to see how he weathers the current onslaught of these accusations. But I would expect that given his position in the polls, he will be a driving component to this primary process.” The dinner was organized by Sen. Saxby Chambliss (Ga.).

House Republicans meeting this afternoon with Cain during a closed-door meet-and-greet at the Capitol Hill Club indicated they might be open to supporting his candidacy, but they said Cain has work to do in convincing Members to support his agenda and the idea that he is presidential. Cain voluntarily raised the issue of the sexual harassment allegations, saying they are false and that he would continue to fight them vigorously, according to Members in the room.

Lost amid the harassment charges is the difficulty Cain might have, if he were to win the GOP nomination, in convincing Republican Members to support his 9-9-9 plan, which would reduce individual and corporate income taxes to a flat 9 percent but also install a national, 9 percent sales tax. Rep. Jeff Denham (R-Calif.) said several Members had questions about the plan and that Cain made the pitch, but the lawmaker indicated that it’s not going to be an easy sell.

“I think he brought some clarity to it. I don’t know that it’s sold yet. I mean you have different states that have different tax policy,” Denham said.

Rep. Marlin Stutzman said Cain has to prove himself worthy of the nomination and capable of serving as president, particularly given his thin political résumé and meteoric rise in the polls. But if Cain was feeling the effects of the three-day-old scandal, he wasn’t showing it, the Indiana Republican said. In response to a reporter’s question, Stutzman said the Georgia businessman did come off as “presidential” material.

“There’s a real vetting process, and he’s going through that now,” Stutzman said. “Whatever the situation with the allegations, he’s going to have to deal with it. And, look, people will decide.”

Chambliss, a fellow Georgia Republican who has known Cain for years, said Tuesday night’s dinner discussion centered mostly on the candidate’s campaign, with little time spent discussing the harassment claims. Chambliss organizes a weekly dinner for a handful of his fellow Senate Republicans and has previously hosted presidential candidates Jon Huntsman and Mitt Romney.

The scandal, Chambliss said, “was part of the discussion, and he didn’t say anything he hasn’t said to you in the press.”

Cain was scheduled to hold a one-on-one meeting this evening with Sen. Jim DeMint (S.C.), an influential conservative in the Palmetto State and nationally. The two had met previously but have not spent much time together. DeMint said the meeting had been scheduled for several weeks and was part of his effort to meet with all of the candidates seeking the GOP presidential nomination.

DeMint expressed sympathy for the position Cain has found himself in since it was reported that two female employees accused him of sexual harassment in the late 1990s, toward the end of his tenure as president and CEO of the National Restaurant Association. The South Carolina Republican, who backed Romney’s 2008 presidential campaign, said he did not expect to endorse a primary candidate until January, if he does so at all.

“It just sounds like something that happens to a lot of folks I knew — pastors, business people — all my life who just had these accusations. There’s not much you can do about it,” DeMint said. “All he can do is tell the truth. I’m just glad it came out early. It’s something that’s likely to go away.”

Cain’s Capitol Hill stops included a meeting in the Rayburn House Office Building with a House Republican health care caucus and a gathering with Georgia’s GOP House delegation at the Republican National Committee. Although Cain addressed the harassment issue with Members, he angrily refused to take questions from reporters. “Don’t even bother asking me all of these other questions that you all are curious about, OK? Don’t even bother,” he said today, according to published reports.

The former CEO of the Godfather’s Pizza restaurant chain, ex-chairman of the Federal Reserve Bank of Kansas City and one-time talk-radio host was still leading his rivals for the Republican nomination today in the RealClearPolitcs average of national polls with 26 percent. However, all of the polls in the average were conducted before the scandal took hold.