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Facing political gridlock and mounting questions on the credibility of his administration’s sequester warnings, President Obama went around Republican leaders Wednesday evening in a bid to connect with rank-and-file lawmakers who’ve worked with Democrats in the past.

The lines of communication between the White House and GOP leaders are not closed. Obama plans to visit Capitol Hill next week to meet with leaders from both chambers.

But the president held a dinner meeting Wednesday with several rank-and-file Senate Republicans. The meeting, days in the making, was held on neutral territory -- a downtown Washington hotel -- the snowstorm blowing through D.C. did not postpone the rendezvous.

"The President greatly enjoyed the dinner and had a good exchange of ideas with the Senators," a senior administration official said.

The guest list included Republican Sens. John McCain of Arizona, Lindsey Graham of South Carolina, Kelly Ayotte of New Hampshire, Pat Toomey of Pennsylvania, Bob Corker of Tennessee, Ron Johnson of Wisconsin, Dan Coats of Indiana, Tom Coburn of Oklahoma, Richard Burr of North Carolina, Mike Johanns of Nebraska, John Hoeven of North Dakota and Saxby Chambliss of Georgia.

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Senate Republican Leader Mitch McConnell claimed he didn’t mind that Obama was going around Republican leaders.

“I expect the president to talk to various members. Frankly, I wish he'd done more of that over the years,” he said. “And he certainly doesn't have to go through me to call my members. And I'm sure he will, and I encourage him to do.”

The meeting comes after Obama, despite escalating warnings about the impact of the sequester, was unable to halt the $85 billion in fiscal 2013 spending cuts that began to take effect Friday.

The president demanded Republicans replace the cuts with a blend of targeted cuts and tax hikes, but Republicans refused and, in the final days before the deadline, claimed the sequester would not necessarily be as bad as the administration claimed.

Republicans have since accused the administration of trying to amplify the impact by taking steps like releasing illegal immigrants from Southwestern jails. But they’ve also ripped the administration for its apparent overstatements -- such as a claim, later shown to be exaggerated, by Education Secretary Arne Duncan that teachers were receiving “pink slips.” One district in West Virginia later revealed that its teachers had been sent transfer notices, but it had nothing to do with the sequester.

Obama last week also claimed Capitol Hill janitors were going to see a pay cut. The White House continued to stand by the claim even after it was revealed the janitors were merely seeing a potential roll-back in overtime.

But the Washington Post fact-checker on Wednesday reported that average overtime last year was $304 -- hardly the kind of money needed to “make ends meet,” as the White House claimed last week. “Clearly, the sequester is hurting segments of the government and will cut the pay of some government workers. It would be better to focus on those people rather than imaginary victims,” the Post wrote.

The dinner Wednesday could touch on a range of topics. It is likely to focus on the current budget debate -- Congress is trying to pass a budget measure to avoid a shutdown at the end of the month, while Obama still wants lawmakers to approve a long-term package to replace the sequester cuts.

But Obama is also still driving for legislation on immigration, gun control and other issues.

The dinner will be followed up by a rare trip by Obama to Capitol Hill. He'll meet there with Senate Republicans next Thursday and hold a separate meeting with House Republicans, although a date for that meeting has yet to be scheduled. The president also will meet with Senate Democrats on Tuesday.

McConnell said the president requested the meeting through his chief of staff. McConnell's office said the president last attended the Senate GOP's policy lunch in May 2010.

"We have numerous challenges facing the country, and Republicans have offered the president serious solutions to shrink Washington spending and grow the economy," McConnell said in a written statement. "And we will have an opportunity to discuss them with the president at the lunch."

House Speaker John Boehner's office said Wednesday that the president had also requested the meeting next week with House Republicans. The White House and Boehner's office were working to schedule that meeting, the speaker's office said.

White House aides say the president's calls with Republicans focus in part on jumpstarting broader budget talks but also on Obama's proposals for overhauling the nation's immigration laws and enacting stricter gun control measures.

"He is reaching out and talking to members about a variety of issues -- not just our fiscal challenges, but certainly the fiscal issues are among the issues he is talking about with lawmakers," White House spokesman Jay Carney said.

Fox News' Wes Barrett and the Associated Press contributed to this report.