Senate negotiators labored late into Saturday over a last-ditch plan to avert the “fiscal cliff,” struggling to resolve key differences over how many wealthy households should face higher income taxes in the new year and how to tax inherited estates.

As the clock ticked toward a Jan. 1 deadline, the halls of the Capitol were dark and silent. The House and the Senate are shuttered until Sunday afternoon, in part to avoid distractions as the talks over averting sharp tax increases for most American taxpayers entered their final hours.

While Senate Majority Leader Harry M. Reid (D-Nev.) monitored developments by telephone, Minority Leader Mitch McConnell (R-Ky.) arrived at the Capitol shortly after noon. Asked whether he and Reid would be able to strike a deal, McConnell smiled and replied: “I hope so.”

As nightfall approached, top Democratic and Republican aides continued shuttling paperwork with the latest proposals back and forth between the two leaders’ offices, about 60 steps apart.

Under negotiation is a deal that would extend George W. Bush-era tax cuts for nearly all taxpayers but increase rates on top earners. It also would extend unemployment benefits set to expire in January for 2 million people and prevent about 30 million Americans from having to pay the alternative minimum tax for the first time.

McConnell left the Capitol shortly before 7 p.m, revealing few details. “We’ve been in discussions all day, and they continue,” he said. He added, “We’ve been trading paper all day and talks continue into the evening.”

Reid and McConnell have set a deadline of about 3 p.m. on Sunday for cinching a deal. That’s when they’re planning to convene caucus meetings of their respective members in separate rooms just off the Senate floor. At that point, the leaders will brief their rank and file on whether there has been significant progress and will determine whether there is enough support to press ahead with a proposal.

“They both know the clock ends Sunday,” said Sen. Mark Begich (D-Alaska).

If all goes according to plan, the leaders would roll out the legislation Sunday night and hold a vote by at least midday Monday, giving the House the rest of New Year’s Eve to consider the measure.

In the House, Speaker John A. Boehner (R-Ohio) huddled Saturday with his senior staff for two hours but remained on the periphery of the negotiations. Passage in the unpredictable chamber is anything but certain. Boehner told President Obama and congressional leaders Friday that he could commit only to considering a Senate-passed bill and suggested that the House may amend that bill and send it back to the Senate.

House consideration of the measure could become another white-knuckle moment. Boehner would like the eventual deal to be passed by a bipartisan coalition that is roughly equally divided between Republicans and Democrats, GOP aides said. Republicans have not supported tax increases since 1990, and conservative activists were already criticizing any deal to raise taxes on the wealthy.

“There is no agreement worthy of the American people to be found in the current Washington charade. Better if everyone went home immediately,” former House speaker Newt Gingrich (R-Ga.) said Saturday in a statement.

Obama said Saturday that he believed Congress could still head off the automatic tax increases set to kick in on Jan. 1 if lawmakers acted fast.

“Leaders in Congress are working on a way to prevent this tax hike on the middle class, and I believe we may be able to reach an agreement that can pass both houses in time,” the president said in his weekly radio address.

Obama, meanwhile, taped an interview that will air Sunday on “Meet the Press” aimed at keeping pressure on Congress to act. His appearance on the NBC News show is just his second as president.

The first came in 2009 at the height of the debate over Obama’s initiative to expand health coverage for the uninsured. The fiscal-cliff measure under construction in the Senate, by contrast, is more modest, designed to fix none of the nation’s biggest problems.

If an agreement is reached, taxes would go up a bit for the very wealthy. But there would be no changes to entitlement programs, such as Medicare and Social Security, and no grand strategy to raise more cash through a simpler tax code. Instead of a fresh start for the new Congress and Obama’s second term, Washington is likely to be mired in the battle over the budget for months.

“A small deal solves virtually nothing,” said Maya MacGuineas, president of the bipartisan Committee for a Responsible Federal Budget and the driving force behind the multimillion-dollar Fix the Debt campaign. “We still have to find ways to cut spending, reform entitlements, raise more revenues and get the debt under control.”

Next up is a debate over the federal debt limit, which is $16.4 trillion. With the debt set to hit the limit on Monday, Treasury Secretary Timothy F. Geithner has signaled that he can juggle the books for about two months before the nation runs out of cash to pay its bills.

Republicans have vowed to use the debt limit fight, as they did in the summer of 2011, to demand deep federal budget cuts, this time focused on Medicare and Medicaid, the federal health programs whose costs are soaring as the population ages. But with the fiscal-cliff talks likely to produce less than half the new tax revenue Obama is seeking, Democrats are likely to demand additional tax increases as well.

That sets the stage for more of the same in Washington, with uncertainty over fiscal policy hampering economic growth and taking energy away from other critical issues, such as immigration and gun control.

“Instead of having the wind at our back and this in our rearview mirror going into this new year, it’s very apparent that’s not what’s going to happen,” said Sen. Bob Corker (R-Tenn.). “Due again to a lack of courage, we’re going to end up in a situation where our nation is hurt and all those who depend upon a good economy for jobs for their families are going to be damaged.”

In his weekly radio address, Obama seemed to lament the lost opportunity, saying he had put forward “a balanced plan” that would “grow the economy and shrink our deficits.”

“I’ll keep working with anybody who’s serious about getting a comprehensive plan like this done — because it’s the right thing to do for our economic growth,” he said.

But for now, he said, the focus has to be on solving the immediate crisis of tax increases for virtually every American that would “hurt middle-class families, and it would hurt the businesses that depend on your spending.”

Negotiators remained in their Capitol offices long past sundown Saturday trying to seal a bipartisan deal that would preserve tax cuts adopted during the Bush administration for the vast majority of Americans, adding at least $3.5 trillion to the federal debt over the next decade.

Negotiations were focused on whether the Bush tax cuts should be permitted to lapse on income over $250,000 a year, as Obama vowed during his campaign, or on income over $400,000, a threshold that would affect only the top tax bracket and many fewer households.

Senate Republicans have signaled support in recent days for the higher threshold, which Obama endorsed in failed negotiations with Boehner earlier this month. Compared with GOP plans to extend the Bush reductions for all taxpayers, letting them lapse on income over $400,000 would raise about $600 billion over the next decade, according to White House estimates — barely a third of the $1.6 trillion Obama initially hoped to extract from high-income households to help reduce record budget deficits.

Negotiators were trying to resolve a dispute over the estate tax, a critical issue for Republicans who have dubbed it the “death tax” and argue that it punishes people who build successful businesses and family farms.

In an agreement brokered between McConnell and the White House in 2010, estates worth more than $5 million are exempted and taxed above that amount at 35 percent. Republicans want to maintain that structure, while Democrats want to drop the exemption to $3.5 million and raise the rate on larger estates to 45 percent.

Aides close to the talks predicted that the two sides might split the difference, with Democrats swapping the estate tax for the $250,000 threshold.

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