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With federal budget cuts beginning to take effect, House Speaker John A. Boehner on Sunday reinforced his opposition to any deal to reverse the cuts that includes new revenues. But he and senior White House officials left open a narrow path to a comprehensive budget agreement that could restore at least some of the money at some point.

Mr. Boehner, appearing on the NBC program "Meet The Press," said President Obama has already raised taxes nearly $1 trillion to finance his health care program and in January won $650 billion in steeper taxes on high incomes.

"How much more does he want?" Mr. Boehner asked. "When is the president going to address the spending side of this?"

But the speaker offered some hope that the budget process – which begins this week with likely House passage of a spending measure, or "continuing resolution," to keep the government open for the rest of the year — could still end in a comprehensive agreement that lowers the deficit, overhauls the tax code and undoes at least some of the automatic spending cuts.

"I don't think anyone quite understands how it gets resolved," he conceded. "After we do our continuing resolution, we'll begin to work on our budget. The House has done a budget every year that I've been speaker. The Senate hasn't done a budget for four years. They've committed to do a budget this year. And I hope that they do. And out of that discussion and out of that process, maybe we can find a way to deal with our long-term spending problem."

To that end, Mr. Obama made a round of phone calls to senators of both parties Saturday to prod bipartisan deficit talks, Gene Sperling, director of the White House National Economic Council, said Sunday, also on "Meet The Press."

The president's path to a less arbitrary approach to deficit reduction is narrow and uphill, lawmakers of both parties say. House Republican leaders are adamant that they want no changes to the tax code – including closing loopholes that both sides agree are wasteful – as part of a deal to undo the spending cuts. Instead, they say that any closing of loopholes or changes to the tax code should be part of a comprehensive effort to simplify the tax code and lower tax rates but keep the level of revenue received by the government the same.

"Republicans want tax reform. We want to bring rates down for all Americans so that we've got a fairer tax code," Mr. Boehner said. "But to arbitrarily pull out a couple of tax expenditures and to say, 'Well, we ought to use that to get rid of the sequester.' Listen, every American knows Washington has a spending problem."

The Senate's top two Republican leaders, Mitch McConnell of Kentucky and John Cornyn of Texas, are increasingly reflecting that same position, and both fear a primary challenge from their right flank as they move toward their re-election campaigns next year.

Mr. McConnell, appearing on the CNN program "State of the Union," gave no indication that he was willing to undo the automatic cuts with new revenue, only with spending cuts in other parts of the budget.

"We think it's important to the American people to keep this commitment. We're willing to do it. We're willing to talk about reconfiguring the same spending over the next several months," he said.

The White House is left to hope that rank-and-file senators can build momentum for a bipartisan deficit deal that isolates Republican leaders and ultimately raises pressure that cannot be resisted.

"I have talked to a number of colleagues on both sides of the aisle, many of whom are disgusted with the current state of affairs," Senator Susan Collins, a moderate Republican from Maine, said in an interview Friday.

Ms. Collins has already begun work with Senator Mark Udall, Democrat of Virginia, on legislation to mitigate the impact of the across-the-board cuts by giving the Obama administration more latitude to spread them around. She said those efforts could put lawmakers on a path toward a consensus on reversing the cuts as they build in force through this fiscal year, which ends on Sept. 30.

But, she said, the president has to speak frankly to the public about the tough changes that are needed over the long term to Social Security and Medicare as the population ages and entitlement costs continue to rise. It is not enough to keep repeating, as Mr. Sperling did on Sunday, that Mr. Obama is prepared to compromise, she said.

"The president has to go first with plans for Medicare and Social Security," she said. "Then I think you will see more receptivity on the Republican side to an overhaul of the tax code" that raises more revenue.

Senator Lindsey Graham, Republican of South Carolina, reiterated his support for up to $600 billion in new revenue through tax code changes over 10 years in a deficit plan that addresses Medicare and Social Security.

Senator Kelly Ayotte, Republican of New Hampshire, said Sunday on the ABC program "This Week" that she too would be open to more revenues in a broad, bipartisan deal that dealt with entitlements.

Mr. Sperling said Sunday that the speaker had already offered $1 trillion in additional revenue over 10 years in backroom talks with the president in December. About $600 billion in tax increases took effect in January.

"If he was keeping his offer on the table he would at least be willing to consider $400 billion in more revenues," he said.