Mr. Obama has invited about a dozen Republican senators out to dinner on Wednesday night, after speaking with several of them by phone in recent days, according to people familiar with the invitation. And next week, according to those people and others who did not want to be identified, he will make a rare foray to Capitol Hill to meet separately with the Republican and Democratic caucuses in both the Democratic-controlled Senate and the Republican-controlled House.

Since the weekend, the president has called at least a half-dozen Republican lawmakers, mostly senators, in a bid to revive talks toward a long-term deficit-reduction agreement and to press for action on other issues, including immigration, gun safety and climate measures. His calls began after indiscriminate government spending cuts — known as sequestration — took effect on Saturday because Republican leaders and Mr. Obama could not agree on alternative measures.

“Maybe because of sequestration and frustration with the public, the time is right to act, and what I see from the president is probably the most encouraging engagement on a big issue since the early days of his presidency,” said Senator Lindsey Graham, Republican of South Carolina, who received a call from Mr. Obama on Tuesday.

Speaking of the deficit reduction impasse, Mr. Graham added, “He wants to do the big deal.”

Mr. Obama’s call to Mr. Graham followed other conversations with Senators Susan Collins of Maine, Tom Coburn of Oklahoma, Rob Portman of Ohio and Bob Corker of Tennessee, all Republicans. Mr. Corker called his conversation with the president “constructive.”

Mr. Portman, while reluctant to detail private talks, was also positive, saying: “I think there’s a window of opportunity between now and the end of the summer. This is the last best chance.”