Obama says fiscal cliff deal in sight NEW: No House vote Monday night, leadership says NEW: McConnell says a deal is "very, very close," but GOP blasts Obama comments Possible agreement calls for increase in income, estate taxes The fiscal cliff triggers broad tax increases and an automatic $110 billion in spending cuts Go to CNN iReport to share your video. (CNN) -- An agreement to avert the fiscal cliff of automatic tax increases and spending cuts appeared "within sight," President Barack Obama said Monday, but lawmakers said nothing will pass Congress before a midnight deadline. Senators were "very, very close" to a deal, having worked out an agreement on taxes, Senate Minority Leader Mitch McConnell said Monday afternoon. But the House of Representatives won't vote on any plans to avert the fiscal cliff on Monday, leaders have told members. At the White House, Obama said the deal now on the table 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 2 million people, Obama said. But Obama did not sound hopeful a deal was imminent, saying he expected to remain at the White House for New Year's Eve as a midnight deadline neared. "They are close, but they're not there yet," Obama said. And one thing we can count on with respect to this Congress is that if there is even one second left before you have to do what you're supposed to do, they will use that last second. The president urged supporters to "Keep the pressure on over the next 12 hours or so." And he cautioned that even if lawmakers can head off an increase in taxes for middle-class families as the tax cuts enacted under the Bush administration expire, they still have to figure out how to mitigate the possible damage from sharp spending cuts that are scheduled to take effect in 2013. But his chiding of lawmakers grappling for a deal drew anger from Republicans. Sen. Bob Corker, R-Tennessee, called the president's comments "very unbecoming of where we are at this moment" and added, "My heart's still pounding." "I know the president has fun heckling Congress," Corker said. I think he lost probably numbers of votes with what he did. He didn't lose mine. I am not that way," he said. I am going to look at the substance. But it is unfortunate that he doesn't spend as much time working on solving problems as he does with campaigns and pep rallies. There's little practical difference in settling the issue Monday night versus Tuesday, House GOP sources said. But 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. 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. 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