CBS News said it confirmed negotiators had reached an agreement 3 hours before the mandated package of hundreds of billions of dollars in spending cuts and tax hikes were to go into effect.
The network said it was unclear when the Senate would vote, but if it passes the House will vote Tuesday at the earliest. That means technically the nation will go over the cliff but without any impact since it would likely be effective retroactively.
CBS said the agreement calls for current tax rates to be extended for all wage earners making less than $400,000 and couples making less than $450,000.
Automatic spending cuts called for under the sequester will be delayed two months to give Congress and the White House time to come up with a wider-ranging deal on spending cuts.
The estate tax will be set at 40 percent with the first $5 million worth of property, indexed to inflation, exempt. Capital gains and dividend tax rates will rise from 15 percent to 20 percent.
There also will be a permanent fix to the alternative minimum tax, and doctors will be shielded from a reimbursement gap for treating Medicare patients.
Unemployment benefits will stay in place.
The tax credit for renewable energy companies also will be extended for another year.
Roll Call reported Vice President Joe Biden was meeting with Senate Democrats to recommend the package.
Roll Call's sources said the tentative agreement is subject to change.
Earlier Monday President Obama said Democratic and Republican congressional leaders were near a deal to avert the "fiscal cliff" but were not there yet.
"Today, it appears an agreement to prevent a New Year's tax hike is within sight, but it's not done," Obama said during a televised statement. "There are issues left to be resolved, but I am hopeful Congress can get it done."
Obama told the audience it was his preference to solve all the issues as part of a "bigger deal" in a responsible and balanced way.
With this Congress, he said, it was "too much to hope for at this time. Maybe we can do this in stages."
He stressed the threat of tax increases was only one part of the budget package Congress passed last year that became known as the "fiscal cliff," and automatic, across-the-board spending cuts also must be dealt with in a "balanced" manner "as part of the equation."
He said he would be willing to reduce the government's Medicare bills, but that kind of reform "has to go hand in hand" with tax-code reform that closes loopholes for the wealthy and corporations.
"If the Republicans think I will finish the job of deficit reduction through spending cuts alone," Obama said, "they've got another [think] coming."
There will be "shared sacrifice" from all taxpayers "as long as I'm going to be president," Obama said.
On the Senate floor, Sen. John McCain, R-Ariz., chided Obama for "antagonizing" House Republicans as he urged Congress to strike a deal. McCain ran against Obama for president in 2008.
If a last-minute deal had not been struck, Reid was expected to move to bring a vote on a stopgap measure pushed by Obama that would retain lower tax rates for incomes $250,000 or less and extend jobless benefits, The New York Times reported.
If deal hadn't been reached by midnight, tax rates would have gone up for virtually all Americans, 2 million people receiving extended unemployment benefits would have lost their stipends, and severe federal spending cuts would have kicked in, potentially sending the U.S. economy back into recession.
In addition, nearly 30 million taxpayers would have been required to pay the costly alternative minimum tax for the first time.
Biden and Senate Minority Leader Mitch McConnell, R-Ky., conducted talks ahead of the midnight deadline.
Biden, an experienced dealmaker who was a senator for 36 years, entered the talks Sunday at McConnell's request.