Those cuts would be replaced with savings totaling $314 billion, achieved through increases in the amount federal employees contribute toward their pensions and through cuts in social programs such as food stamps and the health care law that Obama signed earlier in his term.
Ironically, the votes were set in motion earlier in the week, after Boehner and Obama had significantly narrowed their differences on a compromise to avoid the fiscal cliff.
Republican officials said that members of the GOP leadership had balked at the terms that were emerging. Democrats said Boehner's abrupt decision to shift to his Plan B — legislation drafted unilaterally by Republicans — reflected a calculation that he lacked support from his own rank and file to win the votes needed for the type of agreement he was negotiating with the president.
Asked at a news conference a few hours before the scheduled vote if that were so, Boehner avoided a direct answer. "Listen, the president knows that I've been able to keep my word on every agreement we've ever made," he said.
At the same time, Boehner hinted broadly that however Democrats end up responding to the legislation he placed before the House, it will not be the end of the attempt to keep the economy from reaching the fiscal cliff.
"Our country faces serious challenges. The president and I in our respective roles have a responsibility to work together to get them resolved. I expect that we'll continue to work together."
Obama made it clear on Wednesday that he, too, is prepared for further negotiations, and numerous officials in both parties in the Senate predicted that might happen quickly after the votes in the House.
The tax bill would prevent scheduled increases from taking effect on Jan. 1 on all income under $1 million. Above that, the current rate of 35 percent would rise to 39.6 percent, the level in effect more than a decade ago when then-President George W. Bush signed tax cuts into law that now are expiring.
The top rates also would rise on capital gains and dividends from 15 percent to 20 percent.
By any measure, the two bills in the House were far removed from the latest offers that officials said Obama and Boehner had tendered.
Obama is now seeking $1.2 trillion in higher tax revenue, down from the $1.6 trillion he initially sought. He also has softened his demand for higher tax rates on household incomes so they would apply to incomes over $400,000 instead of the $250,000 he cited during his successful campaign for a new term.
He also has offered more than $800 billion in spending cuts over a decade, half of it from Medicare and Medicaid, $200 billion from farm and other benefit programs, $100 billion from defense and $100 billion from a broad swath of government accounts ranging from parks to transportation to education.
In a key concession to Republicans, the president also has agreed to slow the rise in cost-of-living increases in Social Security and other benefit programs, at a savings estimated at about $130 billion over a decade.
By contrast, Boehner's most recent offer allowed for about $940 billion in higher taxes over a decade, with higher rates for annual incomes over $1 million.
His latest offer seeks about $1.2 trillion in spending cuts, not counting the change in the cost-of-living adjustment that Obama has said he can accept. He is seeking $600 billion in savings from Medicare and Medicaid, $200 billion from other benefit programs and $300 billion from a range of government accounts.
Associated Press writers Andrew Taylor, Alan Fram and Jim Kuhnhenn contributed to this report.