Without such action, taxes would increase on Jan. 1 for every taxpayer. National polls have shown that most Americans would blame Republicans.

Mr. Obama, backed by Congressional Democrats, is proposing as he has for four years that the Bush tax rates be extended permanently for all income below $250,000 a year. In negotiations with Mr. Boehner he had tentatively agreed to raise that threshold to $400,000, and Congressional Democrats on Friday said they would go as high as $500,000 if it would seal a deal with Republicans.

But the Republicans’ rejection of Mr. Boehner’s bill on Thursday indicated that such a concession by Democrats would not sway the antitax absolutists among them. The speaker’s so-called Plan B would have extended the Bush tax cuts for income up to $1 million, meaning a tax increase for only an estimated 0.3 percent of households, yet that was too much for many of his members.

While the strategy that Mr. Obama and Mr. Reid are now pursuing requires the acquiescence of both Republican leaders, Mr. McConnell has given no public indication whether he would give it. Asked at the Capitol before Mr. Obama’s statement whether he would agree not to filibuster the stripped-down bill, he stepped onto an elevator and said “Merry Christmas.”