The agreement emerged after White House officials gave in on the last contested issue, yielding to GOP wishes on how to handle estate taxes, aides said.
The revelations about the pending deal came after Obama had said a deal was "within sight," and House Republican leaders announced they would hold no votes Monday night, making it appear that the nation would go over the fiscal cliff for at least a day.
The Senate was moving, however, toward a late-night vote on the agreement negotiated by McConnell and Biden. With the House likely to reconvene at noon on Tuesday, a deal appeared imminent that would cancel historic tax hikes for most Americans.
"I think it's highly likely that some time this evening there'll be a vote on the Senate side," Sen. Bob Corker, (R., Tenn.), said as he emerged from a meeting with fellow Republicans and signaled that talks are continuing. "I think they're attempting to get the legislative language in order and vote on it tonight, you know, 1, 2, 3, 4 in the morning, whatever."
The announcement that the GOP-controlled House would not vote on New Year's Eve came after Obama urged lawmakers to "stop taxes going up for middle-class families, starting tomorrow," and he called on them to remain focused on the needs of the American people rather than politics.
In what the White House billed as an event with middle-class Americans, Obama said there were "still issues left to resolve" in the talks. He said he was "hopeful that Congress can get it done. But it's not done."
Obama said the potential agreement that Biden and McConnell were crafting would prevent federal income taxes from rising on middle-class families, extend tax credits for children and college tuition, provide tax breaks to clean-energy companies, and extend unemployment insurance for two million Americans.
He would have preferred to "solve all these problems in the context of a larger agreement," the so-called grand bargain, that would have dealt with spending in a "balanced way," he said.
"But with this Congress, that was obviously a little too much to hope for at this time," Obama said, adding that perhaps "we can do it in stages."
Congressional Republicans immediately pushed back, objecting to comments that one GOP senator described as "heckling Congress."
The president made the remarks as negotiators moved closer to a deal but were still hung up on spending, with Democrats so far resisting Republican proposals for spending cuts that would come in exchange for delaying automatic spending cuts at federal agencies for just three months.
As Obama prepared to deliver remarks about the fiscal cliff at the White House, negotiators for the administration and McConnell appeared to have nailed down many of the most critical tax issues, including a plan to let taxes rise on income over $450,000 a year for couples and $400,000 a year for individuals, according to people in both parties familiar with the talks.
McConnell said after Obama's speech that he and Biden spoke multiple times Monday morning since their first 6:30 a.m. call and managed to resolve their differences on taxes. But he echoed Obama's contention that the two sides had not yet resolved a dispute about whether to delay automatic spending cuts. McConnell urged Congress to pass the tax agreement - and debate replacing the so-called sequester, as the automatic spending cuts are known, in coming months.
"We'll continue to work on finding smarter ways to cut spending, but let's not let that hold up protecting Americans from the tax hike that will take place on New Year's Day," he said. "We can do this. We must do this."
Under the proposed accord being hammered out by Biden and McConnell, households earning less than $450,000 would largely escape higher income tax bills, though couples earning more than $300,000 a year and individuals earning more than $250,000 would lose part of the value of their exemptions and itemized deductions, under the terms of the emerging agreement.
Low-income households would also benefit from a five-year extension of credits for college tuition and the working poor first enacted as part of Obama's stimulus package in 2009. And businesses would see a variety of popular tax breaks extended, including a credit for research and development.
The tax on inherited estates would rise from 35 percent to 40 percent, though Democrats agreed to keep in place the current exemption for estates worth up to $5 million. And nearly 30 million households would be protected from paying the costly alternative minimum tax for the first time - either on their 2012 tax returns or at any time in the future. The developing agreement calls for a permanent fix.
The two sides also appeared to have reached consensus on unemployment benefits, with Republicans acceding to Democratic demands to keep benefits flowing to the long-term unemployed for another year. Medicare payments would not be cut for doctors next year, and the cost of preserving those programs would not be offset with other spending cuts.