As though he needed it, House Speaker John Boehner received yet another reminder Tuesday of his principal challenge of finding a resolution to the fiscal cliff morass without alienating the core of his own party, a position that has been all too familiar in recent years.

Shortly after Republican House leaders offered a proposal to avert impending tax hikes and spending cuts, conservatives attacked it as a betrayal of core principles, putting the top GOP lawmaker in a difficult bargaining position against President Barack Obama.

"Speaker Boehner's $800 billion tax hike will destroy American jobs and allow politicians in Washington to spend even more, while not reducing our $16 trillion debt by a single penny," Republican Sen. Jim DeMint said Tuesday in a statement. The South Carolina senator – an influential figure among conservatives – was referring to the new revenue projected in the GOP proposal that would come from closing loopholes and deductions in the tax code rather than rate increases.

The president continues to push taxes increases for those making more than $250,000 as part of his plan to raise $1.6 trillion over the next 10 years, but he suggested those tax rates could eventually be lowered. NBC's Kristen Welker reports.

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The White House rejected that proposal at first glance, reasoning that Boehner and House Republicans offered no concessions to Obama's central demand that income tax rates be allowed to increase for the wealthiest Americans.

In short, Boehner is being pulled in opposite directions by an administration which demands more compromise from Republicans, and by conservatives who expect the speaker to cede no ground. That leaves him with few options to craft a deal. (Boehner allies point, ironically, to some past comments by DeMint suggesting allowing some tax increases might be politically expedient.)

Tim Phillips, the president of the Koch Brothers-backed group Americans for Prosperity, said Monday that Boehner's proposal "leaves conservatives wanting."

"By placing an $800 billion tax hike on the table, Republican Leaders are engaging in little more than pre-emptive capitulation," Heritage Action -- the political wing of the Heritage Foundation think tank -- said in talking points provided Monday to its followers. "The latest Republican proposal to President Obama is nothing but bad policy and a highly questionable negotiating tactic."

The fiscal cliff counter-offer issued by House Republicans has one thing in common with last week's White House proposal – neither was designed to win any bipartisan support. The Daily Rundown's Chuck Todd reports.

This pressure on Boehner from his right flank could have the unintended benefit of signaling to the White House just how narrow his space to negotiate really is. But if Republican leaders do strike a deal with Obama, rank-and-file GOP lawmakers might publicly break with their leaders for fear of alienating constituents and incurring a primary challenge in 2014.

That’s a now-familiar dynamic to anyone who closely tracked the fiscal fights which dominated Congress for much of 2011. The government was brought to the brink of shutdown several times as conservatives balked at supporting deals Boehner had struck with Obama and the Democratic-held Senate. This same discord produced that summer’s debt limit deal, which established the automatic spending cuts that make up half of the fiscal cliff.

It’s this same cast of characters who must now forge the kind of compromise which has eluded Washington for months.

“We're nowhere. We're farther than where we started,” Wisconsin Rep. Paul Ryan, the former GOP vice presidential nominee, said of the negotiations Tuesday on WTMJ radio in Wisconsin. He said that Obama is now demanding higher tax rates than the ones on which he had campaigned.

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As if to illustrate the delicate balance Boehner must strike, when Oklahoma Republican Rep. Tom Cole suggested last week that Republicans should accede to Obama’s request that Congress authorize extended tax cuts for all but the top 2 percent of earners, the GOP speaker emerged to dismiss it. “I told Tom earlier at our conference meeting that I disagreed with him,” Boehner said.

Adding to Republicans’ political headache was a new poll released Tuesday suggesting Republicans would assume the lion’s share of blame from voters if the government were to cross into the fiscal cliff. Fifty-three percent of Americans said Republicans in Congress would be more to blame for a failure to reach consensus in a Nov. 29-Dec. 2 poll conducted by the Pew Research Center and the Washington Post. That’s unchanged from a month ago, despite a messaging barrage by both GOP leaders and the president over the past few weeks.

“If you watch the nature of what Republicans have done here, we’ve talked about this and passed legislation last year,” House Majority Whip Kevin McCarthy, R-Calif., responded on CNBC. “We’ve responded to every presidential proposal. We’ve been first on the mix. The actions don’t hold up to where the polls are, but we want to make sure we solve this problem and that we don’t go over the fiscal cliff.”

News reported Monday by NBC that Boehner had moved to strip four GOP lawmakers of plum committee assignments due to disloyal behavior during the past two years has only threatened to exacerbate tensions between Boehner and influential conservatives, too.

Alex Wong / Getty Images Speaker of the House Rep. John Boehner, R-Ohio, leaves after a news conference Nov. 30, 2012 on Capitol Hill in Washington, DC.

"The dirty little secret in Congress is that while refusing to kowtow to the wishes of party leaders can sometimes cost you some perks in Washington, the taxpayers back home are grateful," said Club for Growth President Chris Chocola.

"This is a clear attempt on the part of Republican leadership to punish those in Washington who vote the way they promised their constituents they would -- on principle -- instead of mindlessly rubber-stamping trillion dollar deficits and the bankrupting of America," added Matt Kibbe, of the Tea Party-oriented group FreedomWorks.