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But that bipartisanship doesn’t extend to the broader spending cuts included in House Republicans’ $61 billion funding reduction for the remainder of the fiscal year.

Mr. Obama has issued a veto threat on that bill, saying House Republicans’ cuts are unacceptable, and Senate Democrats said the non-spending provisions of the bill — such as restrictions on Planned Parenthood and on Obama administration rules and regulations — will also have to be struck.

Mr. Reid pointed to two local Republican mayors in his home state of Nevada who said their cities would suffer under the cuts.

Administration officials also ramped up their pressure, with Homeland Security Secretary Janet Napolitano warning that proposed cuts to her department will mean even longer waits at the airport for security checks.

And Ben S. Bernanke, chairman of the Federal Reserve, testified to Congress on Wednesday that reducing government spending the way Republicans propose would hurt the economy and could lead to hundreds of thousands fewer jobs than would otherwise exist if Congress kept spending at the same rate as 2010.

In the meantime, Republicans found themselves suffering for their own semantics earlier in the debate.

GOP leaders contended that their $61 billion in cuts was actually $100 billion — which is true only when measured against Mr. Obama’s proposal, which had called for increased spending.

On Wednesday, White House press secretary Jay Carney employed that same formula to argue that the president has already accepted $45 billion in cuts — because current spending is frozen $41 billion below the president’s request and Congress just approved an additional $4 billion in cuts.

“We have come almost halfway already, we have met them halfway, which in many ways is the perfect definition of an attempt to compromise,” he said.

Also Wednesday, the House overwhelmingly passed a bill to extend federal highway spending through Sept. 30. That money had been slated to run out on March 4, but highway spending is popular among lawmakers, and they voted to extend it 421-4, with just three Republicans and one Democrat opposing the measure.