For House Republicans, the maneuvers represented a near-reversal of their original strategy in September of going to the mat over the debt limit but not shutting down the government. Now, under pressure from falling poll numbers and angry business supporters, they are seeking a compromise on the debt ceiling. Yet for now, they are still refusing to finance and reopen the government without some concessions.

Mr. Boehner and his colleagues left the White House without speaking to waiting reporters, and quickly gathered in his Capitol suite for further discussion. Their debt limit proposal could come to a vote as soon as Friday.

Before the White House meeting, administration and Congressional Democrats said they were skeptical that House Republican leaders could pass the proposal. A large faction of Tea Party conservatives campaigned on promises never to vote to increase the nation’s debt limit. And Congressional Democrats vowed to oppose any proposal that did not also fully finance a government now shuttered since the fiscal year began Oct. 1.

“We’ll see what they’re able to pass,” said Mr. Obama’s press secretary, Jay Carney.

Senate Democrats had their own White House meeting with Mr. Obama and Vice President Joseph R. Biden Jr. three hours before the House Republicans arrived, and the majority leader, Senator Harry Reid of Nevada, declined to embrace the Republicans’ debt limit proposal until he saw it. He told reporters that Democrats would not negotiate on further deficit reductions until House Republicans agreed to the measure passed by the Senate to finance and open the government through mid-November.

“Not going to happen,” Mr. Reid said. “Open the government,” he added. “There is so much pain and suffering out there. It is really tear-jerking, to say the least.”

Separately, members of the Senate’s Republican minority, who are to meet with Mr. Obama on Friday, worked on a proposal for full-year government funding — at levels reflecting the across-the-board spending cuts known as sequestration, but giving agencies flexibility to shift money around. They are considering adding it to any short-term debt limit increase that the House might pass and send to the Senate.