UPDATE: President Obama will deliver remarks about the fiscal cliff at 1:30 p.m. ET Monday, according to a release from the White House.
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A possible deal to avert the midnight deadline for the "fiscal cliff" of automatic tax increases and spending cuts began to take shape Monday, including an agreement to raise the income tax rate on top earners to what it was during President Bill Clinton's last term in office, according to sources close to the process.
The potential deal also includes an increase in the estate tax and an extension of unemployment benefits, according to the sources.
Other issues remain in play, and it remains to be seen how the GOP-controlled House, which earlier refused to back a $1 million threshold for higher taxes, would respond to any deal.
But the two sides are closer to an agreement than they were Sunday, the sources told CNN.
The developments allowed a slight sense of optimism to creep into the discussion as Congress worked in a rare New Year's Eve session.
"I think now there's a better than a 50-50 chance that we will avoid the fiscal cliff by midnight tonight," said Rep. Chris Van Hollen of Maryland, the ranking Democrat on the House Budget Committee.
The proposed agreement would raise rates on top earners to Clinton-era levels, which topped out at 39.6% in 2000 before falling to the current 35% under tax cuts championed by President George W. Bush. It was unclear if the proposal would adjust the tax rates for inflation, but sources said it would spare 98% of Americans from any tax increase.
Just who would pay higher taxes remained a moving target Monday.
President Barack Obama and his Democratic allies waded into the debate about the fiscal cliff seeking tax increases on individuals making more than $200,000 and families with incomes above $250,000. He later offered to raise the threshold to $400,000 as part of a larger deal.
Sen. Dick Durbin, D-Illinois, said Republicans have now offered a $450,000 income threshold for individuals and $550,000 for couples. Democrats countered with $360,000 for individuals and $450,000 for couples, he said.
The movement comes a day after negotiations hit a stumbling block over a Republican demand that a deal include changes to how Social Security benefits are adjusted for inflation.
Senate Majority Leader Harry Reid chastised Republicans for putting up the Social Security issue, while Senate Minority Leader Mitch McConnell appealed to Vice President Joe Biden to help "jump-start" negotiations after complaining that he had received no response to an offer he put on the table.
"I want everyone to know I'm willing to get this done, but I need a dance partner," said McConnell, R-Kentucky.
Reid, D-Nevada, had said earlier that McConnell had shown "absolutely good faith" in the talks, but "it's just that we are apart on some pretty big issues."
If nothing gets done before Monday at midnight, when the Bush administration's 2001 and 2003 tax cuts expire, broad tax increases will kick in, as will $110 billion in automatic cuts to domestic and military spending.
The nonpartisan Congressional Budget Office has predicted the combined effect could dampen economic growth by 0.5%, possibly tipping the U.S. economy back into a recession and driving unemployment from its current 7.7% back over 9%.
McConnell and Biden, who served together in the Senate for more than two decades, had a "pretty fruitful" conversation Sunday, said Sen. Bob Corker, R-Tennessee.
Top-level sources on both sides of the negotiations said on condition of anonymity that talks are primarily now in the hands of McConnell and Biden, and they are keeping Reid and House Speaker John Boehner, R-Ohio, informed.
On Sunday night, Boehner met with House GOP leaders and told them to sit tight and stick together as he awaits news on whether the Senate can strike a deal.
After the meeting, Rep. Tom Cole, R-Oklahoma, told reporters that Boehner said: "I've stayed out of those negotiations."
"Every time we get involved in them, we sort of get burned, so we're going to let the Senate work its will, see what they do and what they send us, and we'll act accordingly," he said.
As he headed home Sunday night, Reid was asked about progress, and he said: "Talk to Biden and McConnell." On Monday, McConnell declined to say if he was optimistic.
Obama, meanwhile, laid the blame over the stalemate at the feet of Republicans.
"They say that the biggest priority is making sure that we deal with the deficit in a serious way. But the way they're behaving is that their only priority is making sure that tax breaks for the wealthiest Americans are protected," he said Sunday on NBC's "Meet The Press."
to be their only overriding, unifying theme."
During the interview, Obama said he was willing to consider changing the way inflation is calculated for Social Security benefits, meaning that future Social Security recipients would receive less money over time, even though it was "highly unpopular among Democrats" and opposed by the AARP, the powerful lobby for seniors.
"In pursuit of strengthening Social Security for the long term, I'm willing to make those decisions," Obama said.
"What I'm not willing to do is to have the entire burden of deficit reduction rest on the shoulders of seniors, making students pay higher student loan rates, ruining our capacity to invest in things like basic research that help our economy grow. Those are the things that I'm not willing to do."
But a Democratic source, who did not want to be identified because of the closed nature of the talks, said members understand Obama proposed making inflation adjustments to Social Security benefits as an element of a larger deal that also would change how the federal debt ceiling is adjusted -- an element no longer included in the plans.
Most Democrats oppose the inflation adjustment to Social Security, known as "chained CPI," but many were wiling to go along with it as part of a larger deal, the source said.
Sen. Chuck Schumer, D-New York, told ABC's "This Week" that he thought the chances of a short-term, last-minute deal brokered by Senate leaders were better than 50-50, while Sen. Lindsey Graham, R-South Carolina, told CNN that Obama will probably win the fiscal cliff battle, but it will do little to help the nation's deficit problem.
"The president will get a political victory, a trophy for the president politically, but it will not change our debt situation or reduce our deficit in any meaningful way," Graham said. "It will be a political victory that is hollow in nature when it comes to preventing our country from becoming Greece."
Other Republicans argued Sunday that Obama's plan hasn't done enough to limit spending.
"The president is doing nothing about the addiction that his administration has to spending. He's the spender in chief," Sen. John Barrasso of Wyoming said on CNN's "State of the Union."
CNN's Matt Smith, Jessica Yellin, Dana Bash, Deirdre Walsh, Lisa Desjardins, Ted Barrett and Ashley Killough contributed to this report.