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The White House on Tuesday rejected House Speaker John Boehner's call for a fall-back option to avert the looming fiscal crisis, in a move the speaker's office said "defies common sense."

The feisty exchange came as both sides crept toward a possible middle ground in private talks. But underscoring the differences that remain, the White House resisted Boehner's bid to craft a "Plan B" just in case those talks fail to yield a compromise before a Jan. 1 deadline -- that's when taxes are set to rise on everyone, followed by a wave of spending cuts.

"Every income tax filer in America is going to pay higher rates come Jan. 1 unless Congress acts," Boehner said Tuesday morning, as he introduced his alternative plan.

Boehner called for taxes to rise only on households making more than $1 million. His plan also called for entitlement cuts and a tax reform debate next year.

White House Press Secretary Jay Carney, though, said it "doesn't ask enough of the very wealthiest in taxes and instead shifts the burden to the middle class and seniors," and cannot pass the Senate.

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Within minutes, Boehner spokesman Brendan Buck fired back: "After spending months saying we must ask for more from millionaires and billionaires, how can they reject a plan that does exactly that?"

Buck accused President Obama of "moving the goal posts" and in the process "threatening every American family with higher taxes."

Boehner, with no less than his leadership standing on the line, is trying to walk a tightrope in negotiations as the president demands tax hikes that many Republicans adamantly oppose.

His call for a "Plan B" could be a failsafe option, in case Boehner is unable to secure the votes to pass any future deal struck with the White House. But it also could be aimed at spurring Obama to compromise a bit more.

"The president is not there yet," Boehner said.

A Boehner aide said the speaker is "hopeful an agreement will be reached," but complained that the president's latest counteroffer includes $1.3 trillion in revenue increases and "only" $850 billion in spending cuts.

"The threat of all current tax rates expiring is too great to not have a backup plan," the aide said.

House Republican Leader Eric Cantor, putting pressure on the president, said the "alternative plan" is being discussed in case Obama "cannot come our way."

"He has not come to where he needs to be," Cantor said.

A big challenge for Boehner is whether he can sell any compromise struck with the White House to rank-and-file Republicans. His offer to raise taxes on those making more than $1 million drew the ire of some conservatives, as did talk that he's considering allowing a debt ceiling increase that would prevent a battle over that issue for another year.

Obama, though, is demanding far more. In an offer presented Monday, the president asked for a two-year reprieve on the debt ceiling. He backed off his position that tax rates rise on households making more than $250,000, upping the threshold to $400,000 -- but that's still a bigger swath of taxpayers than Boehner is looking at. Plus, the president continues to call for stimulus spending to be part of the package.

For taxpayers, a host of benefits and tax provisions are on the line. Without a deal, tax rates rise for everyone -- with middle-income families facing a roughly $2,000 increase next year. Capital gains and dividends rates would also go up, and long-term unemployment aid would expire.

With a deal, most Americans would likely be spared major tax increases -- but those making above a certain threshold could still be hit, depending on what both sides agree to. Certain entitlement programs could also be cut, including Medicare and Social Security. Obama has indicated he's willing to agree to smaller cost-of-living increases for Social Security recipients.

In terms of logistics, the House is tentatively planning to take up the "Plan B" proposal later this week, as early as Thursday. They would do it by amending a bill previously passed by the Senate that tracked more with Obama's original offer.

Then, the House would send the bill back to the Senate -- where it would likely sit as a potential last resort in case Obama and Boehner cannot strike a deal.

The Boehner alternative plan was getting mixed reviews Tuesday from Republican House members. But on the Democratic side, leaders expressed swift opposition even before the White House came out against it.

"Speaker Boehner's 'plan B' is the farthest thing from a balanced approach," Senate Majority Leader Harry Reid's office said in a statement. "It will not protect middle class families because it cannot pass both Houses of Congress."

House Democratic Whip Steny Hoyer called it a "political ploy," and said Democratic leaders would urge their members to vote against it.

Pressed Tuesday on what terms he's looking for in talks with Obama, Boehner said a plan that cuts $1 trillion from the budget and raises $1 trillion in revenue would be "balanced."

Fox News' Chad Pergram contributed to this report.