President Obama’s budget director told Congress Wednesday that automatic spending cuts will slash funding for 16,000 school employees, cut the U.S. Border Patrol and kick 100,000 children out of the Head Start program as the White House sought to up the political pain for lawmakers bickering over how to stave off the cuts.

House Republicans offered to cancel their August vacation and stay in town to work on a solution, but Jeffrey Zients, acting director of the Office of Management and Budget, said the administration can’t deal with the Republicans until they agree to raise taxes on the wealthy.

That stance officially ties together the two biggest issues dominating Capitol Hill right now, both of which threaten to bedevil lawmakers through the end of the year: What to do about the expiring George W. Bush-era tax cuts, and about the automatic spending cuts, known in legislative-speak as “sequesters,” both of which kick in at the beginning of January.

“There are five months remaining for Congress to act. What is holding us up right now is the Republican refusal to have the top 2 percent pay their fair share,” Mr. Zients told the Republicans in an acrimonious hearing before the House Armed Services Committee.

Hours later, the House officially rejected that stance in a bipartisan 256-171 vote to extend all of the Bush-era tax cuts for a year. Nineteen Democrats sided with the Republicans.

“This means that as of now, only the House has acted to stop both the tax increase on small businesses that threatens our economy and the defense cuts that threaten our national security,” House Speaker John A. Boehner said after the vote.

Taxes and the automatic spending cuts have dominated talk on Capitol Hill this week as lawmakers rush to finish business ahead of a monthlong vacation.

Mr. Boehner sent a letter to Senate Majority Leader Harry Reid offering to keep the House in session if the two sides could make progress on the sequesters, but Mr. Reid echoed the White House’s stance that no action can happen until Republicans drop their opposition to any tax increases.

That left Mr. Boehner saying that blame for the impasse and the potentially devastating defense spending cuts slated for Jan. 2 belong purely to Democrats.

“Translation: We’re putting this right at your doorstep. You own it. If Harry Reid, Nancy Pelosi and Barack Obama want to own a small-business tax hike and a devastating cut to our national defense, they are now set up perfectly to do so,” the Ohio Republican told his House colleagues.

The fight is raising the heat across Capitol Hill. Mr. Zients‘ testimony, in which he repeatedly blamed GOP opposition to tax increases, was blasted by Republicans on the Armed Services Committee, who said they are used to working with Democrats and the administration in a bipartisan manner on defense issues.

But Democrats said it was rude of Republicans to talk over Mr. Zients and blamed them for the tone.

Rep. Robert E. Andrews, New Jersey Democrat and deficit hawk, said it was time for legislators in both parties to stop saying merely what they would oppose and start describing what they’d be willing to sacrifice.

“We as a group have to move to the next step,” he said. “We can stipulate the fact that we all don’t like this. Now, what are we going to do about it?”

Under last year’s debt-ceiling agreement, the administration is required to cut $110 billion in spending on Jan. 2, split between defense and domestic spending.

Both sides say the cuts will lead to massive layoffs, particularly among defense contractors, who said that under federal law they’ll be required to warn their employees of job losses 60 days ahead of time — which happens to be just days before the election.

The Obama administration issued guidance this week saying those warnings aren’t necessary, but Republicans said the law is clear and accused the president of trying to hide job losses from the public in the run-up to voting.

None of the choices is particularly palatable to a majority in Congress: Raise taxes to cover $110 billion, cut spending elsewhere to make up for the money, or just cancel the cuts altogether — which would mean deepening the deficit and passing on the cost to future taxpayers, with added interest.

Add to that the debate over the Bush tax cuts, where the two sides are equally stalemated.

Republicans’ tax plan, which calls for a one-year extension of all current rates in order to give Congress breathing space for a broader tax overhaul, cleared the House on Wednesday. But last week Senate Democrats passed a version that allows tax rates to rise for households making $250,000 or more.

The Senate vote was 51-48, barely squeaking by with no Republican support.

That compared to the House GOP proposal, which passed easily and garnered support of 19 Democrats from conservative-leaning parts of the country. One Republican, Rep. Timothy V. Johnson of Illinois, voted against his party.

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