"This agreement is not subject to being reopened," said Senate Republican leader Mitch McConnell of Kentucky. "In other words, we have an understanding."
President Barack Obama announced the agreement a week ago, saying it had been worked out with leading lawmakers in both parties and was essential to help strengthen an economy still struggling to recover from the worst recession in decades.
There is little disagreement among lawmakers about many elements of the bill, but a provision to benefit the wealthy by easing the estate tax draws strong opposition from House liberals.
The Democratic rank-and-file voted last week not to permit the bill to come to a vote unless it had been changed first. They appear to have dropped their ultimatum, and held an evening meeting to consider other possibilities.
The measure commands strong bipartisan support in the Senate, which voted 83-15 on Monday to advance it to a final vote that is expected on Wednesday.
At its core, the bill would extend tax cuts at all income levels that were enacted during the presidency of George W. Bush. It also would renew a program of jobless benefits for the long-term unemployed that is due to lapse, and enact a one-year cut in Social Security taxes. The bill's cost, $858 billion, would be added to the deficit.
At the insistence of Republicans, the plan includes a more generous estate tax provision: The first $10 million of a couple's estate could pass to heirs without taxation. The balance would be subject to a 35 percent tax rate.
The lower estate tax infuriated some Democrats who were already unhappy with Obama for agreeing to extend tax cuts for individuals making more than $200,000 and couples making more than $250,000.
The estate tax was repealed for 2010. But under current law, it is scheduled to return next year with a top rate of 55 percent on the portion estates above $1 million - $2 million for couples.
House Democratic leaders want to bring back the 2009 estate tax levels. That year, individuals could pass $3.5 million to their heirs, tax-free. Couples could pass $7 million, with a little tax planning, and the balance was taxed at a top rate of 45 percent.
Despite the maneuvering, House Majority Leader Steny Hoyer told reporters, "I think, frankly, that ultimately we will pass legislation."
"The vote in the Senate indicates an urgency that is felt by a broad spectrum that the middle-income taxes not be increased come Jan. 1. In order to affect that, you've got to pass the bill," he said.