WASHINGTON — President Barack Obama and Senate leaders were on the verge of an agreement Friday that would let taxes rise on the wealthiest households while protecting the vast majority of Americans from historic tax hikes set to hit in January.

The development marked a breakthrough after weeks of paralysis. After meeting with Obama at the White House, Senate Majority Leader Harry Reid, D-Nev., and Minority Leader Mitch McConnell, R-Ky., said they would work through the weekend in hopes of drafting a “fiscal cliff” package they could present to their colleagues on Sunday afternoon.

As the Senate began haggling over critical details, the emerging deal faced an uncertain fate in the House, where Speaker John Boehner, R-Ohio, failed just one week ago to persuade his adamantly anti-tax caucus to let taxes rise even for millionaires.

Still, on Friday, Obama pronounced himself “modestly optimistic” at a brief news conference at the White House. The ordinarily dour McConnell said he was “hopeful and optimistic.” And Reid immediately began preparing Senate Democrats for what could be a difficult vote.

“Whatever we come up with is going to be imperfect. Some people aren’t going to like it. Some people will like it less,” Reid said on the Senate floor before dozens of silent and attentive senators from both parties. But “we’re going to do the best we can for . . . the country that’s waiting for us to make a decision.”

According to people briefed on the talks, the developing package would protect nearly 30 million taxpayers from paying the alternative minimum tax for the first time and keep unemployment benefits flowing to 2 million people who otherwise would be cut off in January.

The deal also would likely halt a steep cut in Medicare reimbursements set to hit doctors in January and preserve popular tax breaks for both businesses and individuals, such as those for research and college tuition.

Obama has proposed letting tax rates rise on income over $250,000 a year. Senate Republicans have in recent days expressed interest in a compromise that would lift that threshold to $400,000 a year, an offer Obama made to Boehner before the speaker abruptly broke off negotiations last week.

In addition to its political appeal, the $400,000 threshold has practical benefits, Republican aides said: It would limit tax increases to the very top tax bracket rather than the top two brackets. And it would avoid a quirk of the tax code that would cause rates to rise more dramatically for those earning between $250,000 and $400,000 than for households with much larger incomes.

There was also no agreement on how to handle roughly $100 billion in automatic spending cuts that are scheduled for the Pentagon and other federal departments in the fiscal year that ends in September.