Democrats in the White House and Congress also say that they believe Mr. Boehner does hold greater sway among Republican colleagues than he did in the summer of 2011, his first year as speaker, given the chastening experience for junior Republicans of both last year’s budget fights and the 2012 election results.

Contributing to that sense of Mr. Boehner’s greater empowerment was the letter he sent to Mr. Obama last week, in which he acknowledged Republicans’ willingness to raise new revenues as part of a deal: It was also signed by the House Republican leadership team, including Mr. Boehner’s occasional intraparty rival, Representative Eric Cantor of Virginia, the majority leader, and Representative Paul D. Ryan of Wisconsin, the House Budget Committee chairman and former vice-presidential nominee who has a following among antitax conservatives…

Conservative Republicans have been growing increasingly nervous about any deal with Mr. Obama. “I have great respect for the speaker,” said Senator Jeff Sessions of Alabama, the senior Republican on the Senate Budget Committee, “but he doesn’t have my proxy.”