WASHINGTON — They had 80 hours to finish or fail.

Stuck in a “fiscal cliff” stalemate, trust nearing tatters, President Barack Obama and Senate Republicans changed the game after Christmas. It took the rekindling of an old friendship between Vice President Joe Biden and GOP Sen. Mitch McConnell, an extraordinary flurry of secret offers, a pre-dawn Senate vote on New Year’s Day and the legislative muscling that defines Washington on deadline.

Yet the fate of the final agreement remains in doubt as House Republicans show signs of rebellion against the plan.

How the final days of private negotiations pulled the country — maybe only temporarily — back from the precipice of the fiscal cliff marked a rare moment of bipartisanship for a divided government. Several officials familiar with talks requested anonymity to discuss them because they were not authorized to discuss the private details publicly.

Obama, having cut short his Christmas vacation in Hawaii, huddled with congressional leaders Friday afternoon at the White House. Talks between the president and House Speaker John Boehner had failed, so Obama put the fate of the fiscal cliff in the hands of McConnell and Senate Majority Leader Harry Reid.

McConnell made the first move. The Kentucky Republican proposed a plan late Friday night that would extend tax cuts expiring Jan. 1 on family income above $750,000 a year, according to officials. He also wanted to keep tax rates on wealthy estates at 35 percent, slow the growth of Social Security cost-of-living increases, and pay for an offset of the sequester — Congress’s term for across-the-board spending cuts __ by means-testing Medicare. His offer did not include the extension of unemployment benefits Obama had demanded.

Democrats balked and began preparing a counteroffer. It called for extending tax cuts for family income up to $350,000, a concession from Obama’s campaign pledge to cap the threshold at $250,000. The Democratic leader also insisted that any deal include a way to deal with the sequester, plus an extension of the jobless benefits for 2 million Americans.

The negotiating teams traded ideas back and forth on a wintry Saturday. Shortly before 7 p.m., McConnell presented another offer. He dropped the tax cut threshold to $550,000, put the sequester on the table, and offered a one-year extension of the jobless benefits as long as they were paid for through Social Security savings.

Rather than make a counteroffer, the Senate Democratic negotiating team said it was going home for the night.

They reconvened Sunday morning — less than two days before the combination of tax hikes and spending cuts were due to kick in — but still had nothing new to present to McConnell.

Reid’s inaction, officials said, was due in part to McConnell’s insistence on keeping the tax cut threshold above $500,000, a level Obama refused to agree to.

A frustrated McConnell felt he had one last option. He called Biden, his longtime Senate colleague and frequent negotiating partner, and implored him to step in. Seeking to up the pressure on the White House, McConnell publicly announced that he was reaching out to Biden during remarks from the Senate floor during the rare Sunday session.

Until this late stage, Biden had played a secondary role in the “fiscal cliff” talks. He spent Saturday at his home in Wilmington, Del., and was planning to travel to the Caribbean island of St. Croix with his family for the New Year’s holiday.

Obama and Reid both agreed that Biden, a 36-year veteran of the Senate, should take the lead. And once he did, negotiations with McConnell rapidly accelerated.

Around 8 p.m. Sunday, Obama, Biden and staffers met in the Oval Office to discuss what the vice president would deliver to McConnell as the administration’s final offer.

The president set the upper limit for the tax cut extension at family income of $450,000. The sequester must be dealt with, he said, and any delay must be offset through a combination of spending cuts and revenue increases. And Obama demanded that the jobless benefits be extended for one year without a way to make up the $30 billion cost.

Shortly after midnight, Biden had McConnell’s consent on nearly all of the outstanding issues. Only the sequester was unresolved, though both men were open to a plan that called for a separate vote on the sequester, pending Reid’s consent.

The president, vice president and senior staff met in the Oval Office until 2 a.m. to talk through the final details. Obama’s legislative director, Rob Nabors, then headed to Capitol Hill to join Senate negotiators in drafting the outlines of a bill that could be moved on quickly Monday.

Nabors worked continuously throughout the final stages, stopping at home only to change his shirt after leaving the Capitol and before heading to the White House before 7 a.m. Monday.

By then, the White House had spoken to Reid, who rejected the notion of holding a separate sequester vote. Biden broke the news to McConnell in a pre-dawn phone call.

The sequester remained the sticking point throughout Monday, with Biden trading proposals with McConnell’s office for much of the day.

By early evening, discussions coalesced around delaying the automatic spending cuts by two months and paying for the move through a combination of new spending cuts and revenue increases. A hiccup over the estate tax was also resolved.

Shortly before 9 p.m., with three hours until the deadline, Biden and McConnell agreed to the final deal. After Obama called Reid and Pelosi to get their sign-off, the vice president headed up to Capitol Hill to sell the bill to Senate Democrats.

Lawmakers and staff rang in the new year in cramped offices in the West Wing and on Capitol Hill, surrounded by empty pizza boxes and the stray bottle of cheap champagne.

Midnight also marked the moment the government technically went over the “fiscal cliff,” although financial markets were closed Tuesday for the holiday. But optimism ran high on both ends of Pennsylvania Avenue that the impact would be negligible.

The Senate overwhelming approved the Biden-McConnell deal in the early hours of Tuesday morning and sent the bill to the House for final approval.