Obama’s efforts are aimed at jumpstarting budget talks and rallying support for his proposals on immigration and gun control.

The president’s new charm offensive underscores the limitations of his earlier attempts to use public pressure, rather than direct engagement, to win Republican cooperation. That strategy proved futile in recent weeks, as the White House and Congress failed to prevent $85 billion in automatic budget cuts that both sides said they wanted to avoid.

As that “sequester” has started taking effect, Obama has begun quietly calling congressional Republicans to discuss the prospects for an elusive longer-term deficit reduction deal as well as his other second-term priorities. Aides say Obama is concentrating his outreach on lawmakers with a history of bipartisan deal-making and those who have indicated some willingness to support increased tax revenue as part of a big deficit-cutting package.

In both his calls and dinner invitations, the president pointedly has skipped over Sen. Mitch McConnell and House Speaker John Boehner, the GOP leaders who insist that Obama will get no further tax hikes from Capitol Hill.

Republicans have had mixed reactions to the outreach from the president, who previously has shown little appetite for personal engagement with lawmakers, often preferring to assign those efforts to his staff and Vice President Joe Biden.

“He’s never spent anytime reaching out,” said Sen. Tom Coburn, R-Okla., who spoke with the president this week about gun legislation. “The question is, is it starting to change because there is bad poll numbers or is it because he really decided he’s going to lead and solve some of the problems of the country?”

Sen. Lindsey Graham, a frequent critic of the White House on national security issues, said he was encouraged by Obama’s efforts.

“This is how you solve hard problems,” the South Carolina Republican said.

It was during a phone call with Graham this week that the president raised the prospect of a group dinner with Republican lawmakers, an Obama aide said. Graham agreed to put together a guest list.

Along with Graham and Coburn, the lawmakers at Wednesday’s dinner were Sens. John McCain, Kelly Ayotte, Pat Toomey, Bob Corker, Ron Johnson, Saxby Chambliss, John Hoeven, Dan Coats, Richard Burr and Mike Johanns. The two-hour dinner took place on neutral territory – the Jefferson Hotel, a few blocks from the White House.

A White House official said Obama had a good exchange of ideas with the lawmakers, but offered no specifics on what was discussed. The president picked up the tab for dinner, the White House said.

McCain, responding to a reporter’s question about how the dinner went, jokingly said “terrible,” then added that the meal went “just fine.”

Obama has often scoffed at the notion that calling or meeting with Republicans more frequently would soften the ground for substantive negotiations on fiscal issues and other matters.

“I think a lot of folks say, ‘Well, if we look like we’re being too cooperative or too chummy with the president that might cause us problems,’” Obama said, referring to the Republicans, in January. “‘That might be an excuse for us to get a challenge from somebody in a primary.’”

The Republicans joining Obama for dinner may be less concerned with the political implications of sitting down with the Democratic president. Only Graham faces re-election next year.

Obama advisers say they’re hopeful that without the heightened pressure of an imminent fiscal deadline, the president and Republicans can have constructive conversations on a broad deficit-reduction bill that would include concessions from the GOP on tax increases and from Democrats on entitlements.

But unless Boehner and McConnell bend on taxes, prospects for a sweeping deficit deal remain dim.

“You can’t get around the leadership,” said Patrick Griffin, who served as White House legislative director in the Clinton administration. “It’s all about what happens going forward. Are the larger political dynamics going to change enough that Boehner and McConnell see it in their self-interest to change the way they position this?”

There’s also no guarantee Obama and lawmakers won’t find themselves facing a fiscal crisis in the coming months. The Senate still has to pass a bill funding the government after March 27 – the House passed its version of the measure Wednesday – and lawmakers will have to decide whether to raise the nation’s debt limit in May.

Longer term, Rep. Paul Ryan previewed a 10-year plan on Wednesday that he said would eliminate federal deficits without raising taxes. That would tend to continue the budget standoff between the Republicans and Obama, who wants increased tax revenue to be part of any deal. But Ryan, the GOP vice presidential candidate in 2012, held out hope for communication across party lines.

The Wisconsin congressman, who also has spoken with Obama in recent days, said that “we’re going to have to talk to each other to get an agreement about how to delay a debt crisis, how to save this country from a fiscal train wreck that’s coming.”

The president will have an opportunity to make his case to GOP leaders next week when he heads to Capitol Hill for separate meetings with the House and Senate Republican conferences. McConnell announced that Obama would attend the GOP Senate policy lunch, while Boehner’s office said it was still working on a date.

Obama will also meet on Capitol Hill next week with House and Senate Democrats. The White House says all of the meetings were scheduled at the president’s request.

White House aides said that while Wednesday’s dinner would focus more narrowly on budget issues, the agenda for the lunches will be broader and will include discussions on immigration and gun control.

Even as Obama steps up his engagement with lawmakers, aides say he’ll keep trying to build public support for his agenda and continues to believe pressure from the American people can force Republicans into action. Organizing for Action, a group run by former Obama campaign officials, sent an email Wednesday blaming “Republican obstructionism” for the sequester and urging supporters to sign a petition calling on Congress to back the president’s approach for offsetting the cuts.

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Associated Press writers Donna Cassata, Jim Kuhnhenn, Josh Lederman and Erica Werner contributed to this report.

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Copyright 2013 The Associated Press.