“There was enough concern,” Mr. Coats said, “and maybe some legitimate need to do some extension of unemployment benefits, that it shouldn’t have been just shut down.”

Mr. Obama, accompanied by unemployed Americans as he spoke in the East Room of the White House, tried to keep the pressure on congressional Republicans. “We’ve got to get this across the finish line without obstruction or delay,” he said.

But Speaker John A. Boehner of Ohio made clear that in addition to demanding that an extension of expired benefits be paid for, he would also tie it to Republican priorities like building the Keystone XL oil pipeline, expanding exemptions from the Affordable Care Act and opening energy exploration on federal land.

“One month ago, I personally told the White House that another extension of temporary emergency unemployment benefits should not only be paid for but include something to help put people back to work,” Mr. Boehner said after the Senate vote. “To date, the president has offered no such plan. If he does, I’ll be happy to discuss it. But right now the House is going to remain focused on growing the economy and giving America’s unemployed the independence that only comes from finding a good job.”

The White House has labored since 2011 under the belief that bipartisan support in the Democratic-controlled Senate would steamroll the Republican House. That has not worked on an immigration overhaul, a broad transportation bill or a measure outlawing workplace discrimination based on sexual orientation and gender identity. But it has worked on narrower bills like the reauthorization and expansion of the Violence Against Women Act.

Republican leaders accused Democrats of manufacturing a political issue, hoping that the unemployment bill fails at Republican hands. Mr. Schumer said he feared that Republicans allowed the bill to go forward only to steer Democrats into a “cul-de-sac” or a “Mexican standoff” in which each side would offer its own measures to pay for the benefits but neither would compromise.

But leaders in both parties expressed openness to a negotiated settlement, and Democrats gave little indication that they would press for a quick vote. “Now the serious negotiations have to start,” said Senator Jeanne Shaheen, Democrat of New Hampshire.