And both the president and Republicans are committed to raising hundreds of billions of dollars by overhauling the tax code to further limit or end the tax breaks that high-income taxpayers can claim, though they differ in how to do that.

Republicans want to raise all $800 billion from overhauling the tax code, erasing tax breaks for high-income households and using the new revenues both to reduce deficits and to lower everyone’s tax rates. But they have not proposed how to do that, and the president insists it cannot be done without hitting middle-income taxpayers.

Mr. Obama has proposed to keep existing tax breaks but to limit the rate of those breaks for people in higher tax brackets to 28 percent, which would raise $584 billion in a decade. He has proposed variations of that proposal for four years, only to be ignored by both parties because of opposition from charitable groups, the housing industry, insurers and others to curbing deductions for charitable giving, mortgage insurance and other purposes.

Yet both parties seem poised to confront that opposition because they want a budget deal to commit Congress and the White House to overhaul the tax code next year. That is another reason Mr. Obama wants to have the top rates as high as possible: The lower the rates now, the harder it would be to raise revenues next year in overhauling the code.

Some Republicans inside and outside of Congress agree. “Actually, I would rather see the rates go up than do it the other way because it gives us greater chance to reform the tax code and broaden the base in the future,” Senator Tom Coburn, Republican of Oklahoma, said last week.

Roughly splitting the difference on the top rates — settling at 35 percent and 37 percent — would collect nearly $200 billion over 10 years, under half the amount that would be raised if the rates reverted to Clinton-era levels, according to data from Citizens for Tax Justice and the Institute on Taxation and Economic Policy, research groups that advocate for a progressive tax code.

In the years of debate over the Bush tax cuts, which predates Mr. Obama’s first election, $800 billion has been the rough estimate for how much revenue could be raised in the first decade by ending them for the highest-income 2 percent of taxpayers. But most attention focused on the top rates, which account for half of the revenue equation.