The president and Mr. Boehner were careful with their language and left room for compromise despite their fundamental differences about shifting more of the tax burden to high-income Americans. Mr. Boehner would not be very specific on what his goal might be for raising new federal tax dollars.

“I don’t want to box myself in,” he said. “I don’t want to box anybody else in. I think it’s important for us to come to an agreement with the president. But this is his opportunity to lead.”

The speaker, who has struggled with his more conservative rank and file in the past, said he was confident that he could pass a deal if one was reached with the White House. “When the president and I have been able to come to an agreement, there has been no problem getting it passed here in the House,” he said.

House Republican leadership aides found some positive signals in Mr. Obama’s combative tone. They noted that he never specified he wants tax rates to rise, only that he wants additional revenues generated by taxes on the rich. That would give both sides the latitude to devise a restructured tax code that eliminates or limits tax deductions and credits for the rich — or that follows Mitt Romney’s proposal to cap deductions at a set limit for rich households, though many analysts say that approach alone cannot raise the revenue Democrats want.

Any agreement to avert a fiscal crisis in January, when hundreds of billions of dollars in automatic tax increases and spending cuts kick in, now revolves around the definition of tax increases. Mr. Boehner is holding the line against any increase in tax rates, even for the richest Americans, who currently are in the 35 percent tax bracket. But he is leaving open the possibility of a tax overhaul that raises more revenue than the existing code.

Representative Steny H. Hoyer of Maryland, the No. 2 Democrat, echoed the view at the White House that Americans want to see taxes on the wealthy go up. In a statement after Mr. Obama spoke, Mr. Hoyer said that “on Tuesday, the American people made it clear that they support a balanced approach to bring down deficits and set our nation back on a sound fiscal path — one that does not ask working families and those struggling to get by to bear the burden of deficit reduction, but instead asks the wealthiest Americans to pay their fair share.”