Should he choose to come to an agreement, it is clear he will have the support of Democrats. “These are big national moments,” said Representative Chris Van Hollen of Maryland, the ranking Democrat on the House Budget Committee. “It’s important the speaker put the good of the country above Republican House caucus politics.”

Mr. Van Hollen and other Democratic leaders in the House indicated Friday that they would help Mr. Boehner pass any measure he could put together with President Obama’s blessing.

Republicans fell roughly 20 votes short of those needed to pass Mr. Boehner’s bill Thursday night, which sought to make permanent all Bush-era tax cuts for household incomes under $1 million a year. In spite of Mr. Boehner’s assurances that the bill could pass the House — and the insistence of Representative Eric Cantor of Virginia, the majority leader, that Republicans had the votes — the Republican team charged with rounding up votes could not bring enough members on board, even though many of those who declined to support the measure told Republican leadership aides that they secretly hoped it would pass.

Many of the same members who have long had a difficult relationship with Mr. Boehner — Representatives Tim Huelskamp of Kansas and Justin Amash of Michigan, for example — appeared to fall away from him this week. But even more reliable members, like Representative Marsha Blackburn of Tennessee, who was part of the team charged with drumming up votes, also declined to give her support.