STORY HIGHLIGHTS
- NEW: House GOP sources say they're unlikely to vote on any deal until Tuesday
- Deal to avert "fiscal cliff" is "within sight," President Barack Obama says
- Possible agreement calls for increase in income, estate taxes
- The fiscal cliff triggers broad tax increases and an automatic $110 billion in spending cuts
As the fiscal cliff looms, what's your New Year's message to Washington? Go to CNN iReport to share your video.
(CNN) -- An agreement to avert the "fiscal cliff" of automatic tax increases and spending cuts appears to be "within sight," President Barack Obama said Monday.
The deal would prevent a tax increase for the overwhelming majority of Americans, extend the child tax and tuition credits for families as well as those for clean-energy companies and extend unemployment benefits for two million people, Obama said.
"There are still issues left to resolve, but we are hopeful Congress can get it done," Obama said.
Nevertheless, he did not sound hopeful a deal was imminent, saying he expected to remain at the White House for New Year's Eve while lawmakers used up every last second available to them.
"Keep the pressure on over the next 12 hours or so," he urged supporters. "Let's see if we can get this thing done."
Latest updates: Final fiscal cliff scramble
A GOP source told CNN negotiators are "very close" to a deal. The sticking point is $24 billion in spending cuts being sought by Republicans in place of deeper cuts that would automatically take effect at midnight, according to the source.
"It's like looking under the cushions at this point," the source said. "If we can't find that at this point, we should pack this place up."
A congressional Democratic source agreed that spending cuts are the main hold up now.
But House GOP sources told CNN that they think it increasingly unlikely they'll vote on the proposal before Tuesday.
There's little practical difference in settling the issue Monday night versus Tuesday, the sources said. One exception: if House Republicans approve the bill on Tuesday -- when taxes have technically gone up -- they can argue they've voted for a tax cut to bring rates back down, even after just a few hours, GOP sources said.
That could bring some more Republicans on board, one source said.
"I wouldn't overestimate it, but a handful may be the difference we need," the source said.
Read more: Why your paycheck is getting smaller, no matter what
The proposal under discussion Monday afternoon called for rolling back tax rates on the highest-income earners to Clinton-era levels, increasing the estate tax rate, extending unemployment benefits and potentially putting off the $110 billion in automatic spending cuts called for in the legislation that created the cliff, according to sources close to the process.
A source familiar with the negotiations said the proposal under discussion would generate $600 billion in revenues by ending the Bush-era tax cuts on individuals with incomes above $400,000 and families over $450,000. Their tax rate would be 39.6%, the same as it was in 2000 during President Bill Clinton's presidency. The top income rate is currently 35%.
The deal would also increase the estate tax to 40% from the current 35% level and cap itemized deductions for individuals with incomes above $250,000 and household income over $300,000, the source said.
Read more: What if there's no deal on fiscal cliff
In addition to the tax proposals, also under discussion is a proposal to delay the $110 billion in automatic cuts in domestic and military spending due over the next nine months, a draconian approach called sequester that was created by Congress to address the impact of high deficits and debt on the U.S. economy.
Republicans want a three-month delay while Democrats seek to forestall the cuts by one year, a Democratic source told CNN. Another Democratic source said the proposed three-month delay "can't pass."
Despite Obama's backing, one leading Senate Democrat warned the deal could run into trouble -- not only from House Republicans who have long opposed any tax increase, but also from liberals in the Senate who oppose allowing more high-income households to escape a tax increase.
"No deal is better than a bad deal, and this looks like a very bad deal the way this is shaping up," Sen. Tom Harkin, D-Iowa, said.
Read more: Medicare patients may suffer if country goes over fiscal cliff
If nothing gets done before Monday at midnight, broad taxes hikes will kick in as the Bush-era cuts expire and the deep spending cuts will begin to take hold.
The nonpartisan Congressional Budget Office has predicted the combined effect could dampen economic growth by 0.5%, possibly tipping the U.S. economy into a recession and driving unemployment from its current 7.7% back over 9%.
Graham on fiscal cliff: Obama wins 'hollow' political victory
CNN's Matt Smith, Jessica Yellin, Dana Bash, Deirdre Walsh, Lisa Desjardins, Ted Barrett and Ashley Killough contributed to this report.
Obama: Fiscal cliff deal 'within sight'
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Obama: Fiscal cliff deal 'within sight'
