WASHINGTON — Senate leaders groped for a last-minute compromise Saturday to avoid middle-class tax increases and possibly prevent deep spending cuts at the dawn of the new year as President Barack Obama warned that failure could mean a “self-inflicted wound to the economy.”

Obama chastised lawmakers in his weekly radio and Internet address for waiting until the last minute to try and avoid a “fiscal cliff,” yet said there was still time for an agreement. “We cannot let Washington politics get in the way of America’s progress,” he said as the hurry-up negotiations unfolded.

For all the recent expressions of urgency, bargaining took place by phone, email and paper in a Capitol nearly empty except for tourists. Alone among top lawmakers, Senate Republican leader Mitch McConnell spent the day in his office.

In the Republicans’ weekly address, Sen. Roy Blunt of Missouri cited a readiness to compromise. “Divided government is a good time to solve hard problems — and in the next few days, leaders in Washington have an important responsibility to work together and do just that,” he said.

Even so, there was no guarantee of success, and a dispute over the federal tax on large estates emerged as yet another key sticking point alongside personal income tax rates.

In a blunt challenge to Republicans, Obama said that barring a bipartisan agreement, he expected both houses to vote on his own proposal to block tax increases on all but the wealthy and simultaneously preserve expiring unemployment benefits.

Political calculations mattered as much as deep-seated differences over the issues, as divided government struggled with its first big challenge since the November elections.

Speaker John Boehner remained at arms-length, juggling a desire to avoid the fiscal cliff with his goal of winning another term as speaker when a new Congress convenes next Thursday. Any compromise legislation is certain to include higher tax rates on the wealthy, and the House GOP rank and file rejected the idea when he presented it to them as part of a final attempt to strike a more sweeping agreement with Obama.

Lawmakers have until the new Congress convenes to pass any compromise, and even the calendar mattered. Democrats said they had been told House Republicans might reject a deal until after Jan. 1, to avoid a vote to raise taxes before they had technically gone up and then vote to cut taxes after they had risen.

Nor was any taxpayer likely to feel any adverse impact if legislation is signed and passed into law in the first two or three days of 2013 instead of the final hours of 2012.

Gone was the talk of a grand bargain of spending cuts and additional tax revenue in which the two parties would agree to slash deficits by trillions of dollars over a decade.

Now negotiators had a more cramped goal of preventing additional damage to the economy in the form of higher taxes across the board — with some families facing increases measured in the thousands of dollars — as well as cuts aimed at the Pentagon and hundreds of domestic programs.

Republicans said they were willing to bow to Obama’s call for higher taxes on the wealthy as part of a deal to prevent them from rising on those less well-off.

Democrats said Obama was sticking to his campaign call for tax increases above $250,000 in annual income, even though he said in recent negotiations he said he could accept $400,000. There was no evidence of agreement even at the higher level.

There were indications from Republicans that estate taxes might hold more significance for them than the possibility of higher rates on income.

One senior Republican, Sen. Jon Kyl of Arizona, said late Friday he was  “totally dead set” against Obama’s estate tax proposal, and as if to reinforce the point, Blunt mentioned the issue before any other in his broadcast remarks. “Small businesses and farm families don’t know how to deal with the unfair death tax—a tax that the president and congressional leaders have threatened to expand to include even more family farms and even more small businesses,” he said.

Several officials said Republicans want to leave the tax at 35 percent after exempting the first $5 million in estate value. Officials said the White House wants a 45 percent tax after a $3.5 million exemption. Without any action by Congress, it would climb to a 55 percent tax after a $1 million exemption on Jan. 1.

Democrats stressed their unwillingness to make concessions on both income taxes and the estate tax, and said they hoped Republicans would choose which mattered more to them.

Officials said any compromise was likely to ease the impact of the alternative minimum tax, originally designed to make sure that millionaires did not escape taxation. If left unchanged, it could hit an estimated 28 million households for the first time in 2013, with an average increase of more than $3,000.

Taxes on dividends and capital gains are also involved in the talks, as well as a series of breaks for businesses and others due to expire at the first of the year.

Obama and congressional Democrats are insisting on an extension of long-term unemployment benefits that are expiring for about 2 million jobless individuals.

Leaders in both parties also hope to prevent a 27 percent fee cut from taking effect on Jan. 1 for doctors who treat Medicare patients.

There was also discussion of a short-term extension of expiring farm programs, in part to prevent a spike in milk prices at the first of the year. It wasn’t clear if that was a parallel effort to the cliff talks or had become wrapped into them.

Across-the-board spending cuts that comprise part of the cliff were a different matter.

Republicans say Boehner will insist that they will begin to take effect unless negotiators agreed to offset them with specified savings elsewhere.

That would set the stage for the next round of brinkmanship — a struggle over Republican calls for savings from Medicare, Medicaid and other federal benefit programs.

The Treasury’s ability to borrow is expected to expire in late winter or early spring, and without an increase in the $16.4 trillion limit, the government would face its first-ever default. Republicans have said they will use administration requests for an extension as leverage to win cuts in spending.

Ironically, it was just such a maneuver more than a year ago that set the stage for the current crisis talks over the fiscal cliff.