The House of Representatives is expected to vote on it before Wednesday.
The interim New Year's Eve tax deal negotiated by Biden and Senate Republican Leader Sen. Mitch McConnell of Kentucky would raise income taxes on single earners with annual incomes above $400,000 and married couples with incomes above $450,000.
It also blocks spending cuts for two months, extends unemployment benefits for the long-term jobless, prevents a 27 percent cut in fees for doctors who treat Medicare patients and prevents a spike in milk prices.
As of mid-afternoon Monday, the sticking point involved the "sequester," the cuts to spending - about $100 billion to start in 2013 -- that were mandated by the Budget Control Act which President Barack Obama signed into law last year.
Republicans have signaled they might let the sequester take effect unless it was offset by other spending cuts; the GOP has also said it might accept a delay, but only for a few months.
The Obama administration, however, was pushing for a longer delay in implementing the sequester. Otherwise, the president said, replacing those automatic cuts must be "balanced" -- shorthand for a combination of new taxes and other spending cuts.
Obama tried to push talks over the finish line earlier in the afternoon with a statement from the White House.
"Today, it appears that an agreement to prevent this New Year's tax hike is within sight," the president said at the White House on Monday. "But it's not done."
In the absence of a broader agreement to resolve the sequester, McConnell appeared in the Senate floor to request a vote only on the tax element of the fiscal cliff.
"Let's pass the tax relief portion now," he said. "Let's take what's been agreed to and keep moving."
But it's not clear that Democrats, who were led in negotiations by Vice President Joe Biden, would agree to de-link the tax debate from other fights over the sequester and extending expiring unemployment benefits past Dec. 31.
House Republicans were careful to note that it was still possible for them to add votes late on New Year's Eve. But they also argued that there was no Senate-passed legislation on which they could schedule a vote, making the prospect of avoiding the cliff all the less likely.
Democratic and Republican sources in the House told NBC News that a final vote on any deal would now most likely wait until afternoon on New Year's Day, or even on Jan. 2.
Though Congress could still conceivably act after New Year's to preserve existing tax rates -- thereby limiting any lasting effect on consumers -- their inability to reach an agreement until the very last minute could still threaten to rattle the economy and markets.
The House did act late Sunday, though, to clear the way for emergency consideration of Senate legislation if leaders are able to reach an agreement.
The House Rules Committee convened with the purpose of dispensing with a rule instilled by Republicans in the early days of 2011 to require that legislation be posted online for a full 72 hours before a vote in the House.
GOP leaders had sought that rule to showcase their own transparency, and in reaction to actions by the previous Democratic majority to quickly pass legislation during the health care reform battles of 2010.
Republicans' move to sidestep their own rule underscores the urgency of fiscal cliff talks in the final hours of 2012. There were few ironclad assurances, though, that any Senate agreement would necessarily win the support of the House.
The lurching nature of legislating has been characteristic of the Congress during the last two years, and that's a phenomenon that may well continue into the next Congress, when Democrats will continue to retain control of the Senate, and Republicans will hold a slightly slimmer grasp on the House.
Arkansas's Senior U.S. Senator Mark Pryor issued the following statement after the deal was struck:
I'm pleased Senate Democrats and Republicans were able to prevent our country from going over the fiscal cliff. Tonight the Senate passed a package on a bipartisan basis to prevent tax increases on middle-class Americans, create a permanent AMT patch, fix the estate tax to give Arkansas small businesses and family farms the certainty they need, and provide critical tax credits for hard-working families. That being said, I know some may not agree with all the details of this plan. But that's the art of compromise and consensus building, and I hope we see more of it as
Congress works to improve our economy.