Boehner, R-Ohio, proposed his so-called Plan B on Tuesday, a measure that would block tax cuts from being triggered Jan. 1 on everyone but those whose incomes exceed $1 million. The White House said Boehner’s package did not raise enough revenue from the country’s top earners, would leave too big a deficit-reduction burden on the middle class and omitted tax breaks used by families and businesses.

“The president urges the Republican leadership to work with us to resolve remaining differences and find a reasonable solution to this situation today instead of engaging in political exercises that increase the possibility that taxes go up on every American,” White House Communications Director Dan Pfeiffer said in a written statement.

Boehner is planning a House vote on his proposal on Thursday, hoping that it will raise pressure on President Barack Obama to make concessions as both sides continue reaching for a bipartisan deal on averting the “fiscal cliff.” Without an agreement among lawmakers, hundreds of billions of broad tax increases on nearly all taxpayers and budget-wide spending cuts will be triggered in early January.

Senate Minority Leader Mitch McConnell, R-Ky., also tried raising pressure on Obama, saying the president was risking his chance to strike a major budget deal with the GOP.

“He has a real opportunity to show he can govern,” McConnell said on the Senate floor. “He’s letting that opportunity slip away.”

House GOP leaders are facing a problem with their Plan B strategy — they are laboring to line up enough support for the measure in the face of conservatives reluctant to boost anyone’s taxes. Even if it could survive in the House, Senate Majority Leader Harry Reid, D-Nev., has declared it dead in his chamber and now the White House has promised to veto it should it somehow reach Obama’s desk.

The backup plan by Boehner, R-Ohio, would do nothing to head off deep cuts in defense and domestic programs scheduled to begin taking effect in January. And it contains none of the spending reductions that both President Barack Obama and Boehner have proposed in their efforts to strike a compromise.

“The speaker is trying to get as much leverage as he can to deal with the president,” said Rep. Charles Boustany, R-La., describing the pressure Republicans were hoping it would put on the White House. But he added that he wasn’t sure the plan was the best way to get that leverage.

“I’m still trying to figure that out,” Boustany said.

Others were more supportive. Rep. Patrick Tiberi, R-Ohio, said Plan B would let Republicans vote to protect as many people as possible from tax hikes while leaving Democrats with the blame if it failed.

If the Senate decides not to vote on the House bill or ignores it, “That’s not our problem,” Tiberi said. “The ball’s in Harry Reid’s court.”

Besides letting tax rates rise only on incomes exceeding $1 million, Boehner’s Plan B also would boost the top rate on capital gains and dividends from their current 15 percent to 20 percent for earnings over $1 million, preventing higher increases. It would continue current tax levels on inherited estates — less than Obama wants — and prevent the alternative minimum tax from raising taxes owed by 28 million middle- and upper-class families.

Boehner unveiled his backup plan on Tuesday. He did so even though he and Obama have come tantalizingly close to finding a politically palatable combination of revenue increases and budget savings that could slice around $2 trillion from projected federal deficits over the coming decade.

Both sides say those efforts will continue.

Obama has reduced his demands for tax increases to $1.2 trillion over 10 years, to be imposed on incomes exceeding $400,000 annually. In so doing, the president abandoned his campaign season insistence that he would raise taxes on individuals earning over $200,000 and couples making more than $250,000.

Boehner has boosted his revenue offer to $1 trillion, including raising income tax rates on incomes over $1 million. That is a major concession from the leader of a party that has made opposition to higher rates a fundamental tenet for a quarter century.

“I think it’s a mistake for the Republican Party,” said Rep. Jim Jordan, R-Ohio, a leader of House conservatives. “So that’s what I think a lot of members are struggling with.”

Obama has also departed from his party’s orthodoxy by proposing smaller annual cost-of-living increases for Social Security recipients. The new formula for measuring inflation would affect other benefit programs as well and push more people into higher income tax brackets.

The president’s embrace of a plan that would over time cut Social Security benefits has aroused outcries from liberal groups that defend programs for the elderly and poor and that generally support Democrats.

“I promise you, seniors and their families will notice” the benefit reductions, said Pamela Tainter-Causey, spokeswoman for the National Committee to Preserve Social Security and Medicare, one of scores of groups mobilizing opposition to the proposal.

Obama also was pushing for extended unemployment benefits and billions in spending aimed at stimulating the economy.

The president says his plan would cut spending by $1.2 trillion over a decade, including saving $400 billion from health care benefits that could include Medicare for the elderly and Medicaid for the poor.

Boehner wants deeper Medicare savings. And he says Obama is more unbalanced, seeking $1.3 trillion more in taxes and $930 billion in lowered spending.

Besides lingering disputes with the president, the speaker’s efforts to reach an accord with Obama face another difficulty — uncertain support from Boehner’s own lieutenants. One opponent was the influential Rep. Paul Ryan, R-Wis., the party’s defeated vice presidential nominee, said a House GOP aide. The aide was not authorized to publicly discuss the situation and spoke only on the condition of anonymity.

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By Alan Fram. Associated Press writers Andrew Taylor, Jim Kuhnhenn and Stephen Ohlemacher contributed to this report.