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Afterward, Mr. Graham said the conversation was “incredibly encouraging” and chastised the press for overanalyzing Mr. Obama’s motives, noting, “Everybody wants to, you know, be Dr. Phil about what he’s doing.”

“I can’t think of any major accomplishment in America in the private or public sector where no one ever talked to each other,” he said. “So I want to compliment the president for reaching out. I think he’s doing the right thing.”

Sen. Susan M. Collins of Maine also said Mr. Obama called her Tuesday. She reacted with cautious optimism.

“I think the important thing is, for the first time in a very long time, the president appears to be doing some outreach to both Republicans and Democrats, and that’s long overdue,” she told reporters Wednesday.

But critics argue the president’s new charm offensive will have little — if any — impact on moving the two sides closer together on the issues. Instead, they say the effort is a transparent attempt to bolster Mr. Obama’s sagging poll numbers and respond to accusations in the press that Mr. Obama has failed to engage Republicans, choosing instead to use scare tactics to force more tax increases.

“It’s a reactionary move by Obama to try to put the story away,” said Ron Bonjean, a Republican political consultant who has worked for several party leaders on Capitol Hill. “If these stories weren’t out there, it’s highly unlikely he would be sitting down with Republicans tonight. … This is political theater at its best.”

Democrats gave Mr. Obama credit for circumventing GOP leaders and trying to start a dialogue with rank-and-file Republicans. But they too predicted the president’s outreach efforts would do little to change the budget stalemate.

“In politics, as in life, it’s always better to be talking than not to be talking to the other side — it shows that he’s willing to go the extra mile to demonstrate he wants to get things done,” said Jim Manley, a longtime Democratic operative on Capitol Hill who now works for Quinn Gillespie & Associates. “But anyone who thinks it’s going to change anything needs to have their head examined.”

Even as Mr. Obama was inviting Republicans to dinner, Organizing for America, the campaign committee that has transformed itself into an outside group supporting the administration’s agenda, was sending out emails to supporters blaming Republicans for letting the sequester cuts go into effect “because they simply wouldn’t support closing tax loopholes for millionaires and billionaires — for things like yachts and corporate jets.”

⦁ David Sherfinski contributed to this report.