Vice-President Joe Biden briefed Senate Democrats behind closed doors after Republicans and the White House agreed a plan.
The Senate is expected to vote in the early hours of Tuesday morning, with the House to consider a bill later, reports the BBC.
Tuesday is a public holiday in the US, so no immediate effects will be felt.
Analysts say if the effects of the fiscal cliff are allowed to take hold, they could spark a new US recession.
But if a bill passes both the Senate and the House over the New Year holiday, the impact is likely to be minimal.
After a long day of negotiation on Capitol Hill, signs of a deal emerged hours after the House of Representatives was dismissed for the night.
Earlier, President Barack Obama said a deal to avoid the steep tax rises and spending cuts was “within sight”.
Tax cuts passed during the presidency of George W Bush are due to expire at midnight (05:00 GMT), with rates set to increase for all if no deal is passed.
According to an analysis by the Tax Policy Centre, a family of four making $75,000 would see their taxes rise about $3,300 if that happened.
The eventual deal appeared to extend tax cuts for Americans earning under $400,000 (Â£246,000) – up from the $250,000 level Democrats had originally sought.
A huge spending cut known as the sequester – that would see $1.2tn cut from the federal budget over 10 years – would be deferred for two months, reports said, allowing Congress and the White House to reopen negotiations on a wider deal.
Earlier, Obama said he would had preferred to resolve the fiscal cliff through a “grand bargain” that dealt with both long-term spending and tax issues.
“But with this Congress, that was obviously a little too much to hope for at this time,” he said at the White House.
Amid some Republican disquiet over the president’s barbed remarks, the party’s Senate leader Mitch McConnell quickly calmed proceedings in the afternoon.
He backed the president’s assessment and said a deal should be done quickly.
“Let’s pass the tax relief portion now,” said McConnell, who spent Sunday evening and Monday negotiating with Vice-President Joe Biden.
“Let’s take what’s been agreed to and get moving. We’ll continue to work on finding smarter ways to cut spending.”
As well as the income tax rates and spending cuts, the deal under discussion includes:
â€¢ Tax rates to rise on estate inheritances from 35% to 40% after the first $5m for an individual and $10m for a couple
â€¢ Capital gains taxes – affecting certain income from investments – would rise from 15% to 20%, but less than the 39.6% it would rise under after 1 January.
â€¢ Unemployment benefits would be extended for year, affecting an estimated two million people
â€¢ Extending tax credits that help poorer and middle-class families for five years
Any deal needs to pass the 100-member Senate, which is controlled by Democrats, before heading to the House of Representatives, where Republicans hold the majority.
But no votes have yet been scheduled in either chamber.
Also on Monday, the US Treasury said the federal government had hit its self-imposed borrowing limit, the debt ceiling
Treasury Secretary Timothy Geithner sent a letter to lawmakers informing them that some pension and health benefits would be suspended in order to free up borrowing authority until the end of February.
Obama had asked for an extension of debt ceiling as part of the fiscal cliff negotiations, and correspondents say the issue is the next hurdle between Congress and the president.