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. 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.

Bargaining took place by phone, email and paper in a Capitol nearly empty except for tourists. Alone among top lawmakers, Senate Minority Leader Mitch McConnell (R-Ky.) spent the day in his office.

He and Senate Majority Leader Harry Reid (D-Nev.) were in last-minute talks with the White House, working toward votes that could take place Sunday night by both the Senate and House.

"I am more hopeful than I was yesterday," Rep. Peter King (R-Seaford) said.

In the Republicans' weekly radio 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 estate tax emerged as yet another sticking point alongside 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.

House Speaker John Boehner (R-Ohio) remained at arm's length, juggling a desire to avoid the fiscal cliff with his goal of winning another term as speaker when a new Congress convenes Thursday.

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

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

Democrats said Obama was sticking to his campaign call for tax increases for those earning more than $250,000.

There were indications from Republicans that estate taxes, with which Blunt opened his broadcast, might hold more significance than the possibility of higher rates on income.

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. Taxes on dividends and capital gains are also under discussion.

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