WASHINGTON — With Republican leaders in Congress forswearing budget negotiations over new revenues, President Obama has begun reaching around them to Republican lawmakers with a history of willingness to cut bipartisan deals.

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.