With a Republican effort to raise taxes on millionaires shot down by conservatives in his own party, House Speaker John Boehner could be forgiven his thoughts Friday on how to avoid the fiscal cliff: "God only knows."

With just 10 days before tax rates rise on all Americans and deep mandated spending cuts begin to bite, potentially tipping the US economy back toward recession, the two parties accused one another of failing to make concessions in all-important negotiations.

Chastened Republicans said Friday they were not walking away from dealmaking with President Barack Obama on how to resolve a looming fiscal crisis, even as the House leadership ordered its members home for the Christmas holidays.

"I'm interested in solving the major problems that face our country, and that means House leaders, Senate leaders, and the president are going to continue to have to work together to address those concerns," Boehner said.

He spoke to reporters barely 14 hours after his bid to resolve the tax side of the equation, which he named "Plan B," failed disastrously in the House.

Boehner said he remained committed to a grand bargain with Obama that would reform the US tax code and slash spending, notably in entitlement programs like Medicare.

"How we get there, God only knows," Boehner said.

Obama was due to make a statement on the fiscal cliff at 2200 GMT.

Both parties are scrambling to work out a deal that avoids the so-called "fiscal cliff" and the more than $500 billion in tax hikes and spending cuts that would hit Americans in 2013 alone.

Boehner and Republicans sought to pin the responsibility on Obama and his Democrats.

"I don't want taxes to go up. Republicans don't want taxes to go up," Boehner told reporters in the US Capitol.

"But we only run the House. Democrats continue to run Washington."

Boehner's position appears to have suffered following the conservative revolt in his caucus Thursday, and while his grip on the speakership may no longer be as firm, he said he was concerned less with keeping that job than resolving the crisis.

Boehner's tax bid was part of an overall offer to Obama that would raise some $1 trillion in tax revenue -- mostly through closing loopholes and ending certain deductions -- and another $1 trillion in spending cuts, including slashes to some entitlements like Medicare.

"I told the president on Monday these were my bottom lines," Boehner said.

The White House has described its offer as $1.2 trillion in tax revenues and nearly $1 trillion in spending cuts, although Republicans dispute whether all of the austerity measures are real.

One way to make a tax hike on the wealthy more palatable for Republicans is to not make them vote for one at all.

Should the new year arrive with no deal, taxes rise on everyone. Congress could then technically vote to lower taxes on the middle class, something conservatives might be more willing to do.

Like Boehner, Senate Minority Leader Mitch McConnell sought to shift blame to the White House.

"This isn't John Boehner's problem to solve. He's bent over backwards" to prevent taxes from rising on most Americans, McConnell said on the Senate floor, amid back-and-forth sniping with Senate Majority Leader Harry Reid.

"This is a moment that calls for presidential leadership."

Reid accused Boehner of rushing toward the cliff by "wasting a week on this futile political stunt," referring to the failed Plan B.

"But there is still time for the speaker to hit the brakes," he said. "A comprehensive solution will need to be a bipartisan solution."

Other Democrats too said now was the time to come together to prevent fiscal chaos and global economic uncertainty.

"I've told John Boehner that the Leader (Democrat Nancy Pelosi) and I will work on getting sufficient number of Democrats to join with sufficient number of Republicans to pass a balanced, bipartisan agreement that the president will sign," congressman Steny Hoyer, the number two Democrat in the House, told CNBC.

"You can't do it on a partisan basis. John Boehner showed that last night," he added.

"And just walking away and saying, 'now it's up to the president, and up to the Senate,' is not a helpful stance. We all are in this together."