Washington stumbled closer to the “fiscal cliff” Wednesday as President Obama and congressional Republicans dug in further on their positions on taxes, even as no face-to-face negotiations took place.

With hope fading for a deal before Christmas, House Speaker John A. Boehner (Ohio) told his Republican colleagues to prepare for a holiday season of tense negotiations in Washington.

He told reporters of a “deliberate call” he had with Obama on Tuesday evening after each man rejected the other’s latest offer.

“We spoke honestly and openly about the differences that we face,” Boehner said. “But the president’s calling for $1.4 trillion worth of revenue. That cannot pass the House or the Senate.”

The speaker was referring to Obama’s latest offer, which reduced his demand for $1.6 trillion in new taxes to $1.4 trillion. Republicans, who have offered $800 billion in new revenue generated by a rewrite of the tax code, responded with a proposal that was virtually identical to their opening bid from weeks ago.

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Democrats accused Republicans on Wednesday of seeking a permanent extension of expiring tax breaks for the wealthiest 2 percent of taxpayers, something the president has refused to accept; he won reelection after campaigning to end those tax cuts.

Republicans say their proposal to extend the George W. Bush-era tax cuts for one year is a compromise to give lawmakers enough time to overhaul the tax code. Although the GOP does not want to raise rates on the wealthy, party leaders have offered to extract $800 billion from those taxpayers over the next decade by limiting their deductions and other tax breaks.

Economists have warned that going over the fiscal cliff — the set of sizable tax increases and spending cuts mandated to take place early next year — would throw the nation back into a recession.

Aides said that there were no phone calls between the president and the speaker Wednesday and that the last high-level, face-to-face discussion came Tuesday, when White House legislative affairs director Rob Nabors huddled in the Capitol with senior Boehner aides. No progress was reported.

No more discussions are scheduled.

Rep. Chris Van Hollen (D-Md.) said that the framework of a deal must be in place by the end of this week to ensure its passage by Christmas but that such a timeline is now unlikely. Van Hollen, the ranking Democrat on the House Budget Committee, predicted that conservative opposition to higher taxes will require Boehner to offer legislation that could pass largely on the strength of Democratic votes.

“I think the biggest impediment right now is the speaker’s ability to get a decent number of Republican votes for an agreement that I think most people would agree was a fair agreement,” he told reporters at a Wednesday breakfast hosted by the Christian Science Monitor.

House Republicans have rejected any effort to approve a bill passed by the Senate over the summer that would extend the Bush-era tax cuts for everyone except the top 2 percent of earners. Conservatives in the GOP conference object to any plan to raise tax rates, and Republicans leaving a Wednesday morning meeting with Boehner said the leadership team has not suggested to rank-and-file lawmakers that it will ask them to support a compromise plan with higher rates.

Some GOP political strategists fear that if all tax rates go up in January, Republicans could be blamed — an outcome that would be political anathema for a party whose anti-tax position has been at the core of its identity for more than 30 years.

Ed O’Keefe and Rosalind S. Helderman contributed to this report.