With historic tax increases set to hit virtually every American in five days, President Obama and members of the Senate are headed back to Washington on Thursday to take one last shot at a deal to protect taxpayers and the gathering economic recovery.

If anything, hope for success appeared to have dimmed over the Christmas holiday. The Republican-controlled House last week abdicated responsibility for resolving the crisis, leaving all eyes on the Senate. But senior aides in both parties said Majority Leader Harry M. Reid (D-Nev.) and Minority Leader Mitch McConnell (R-Ky.) have not met or even spoken since leaving town for the weekend.

One small sign of progress came from House GOP leaders, who vowed Wednesday to call the House into session and stage a vote on anything the Democratic-controlled Senate approved.

But Speaker John A. Boehner (Ohio) and other top Republicans also demanded that Senate leaders take up a bill approved by the House months ago that would preserve expiring tax cuts for the wealthy as well as the middle class.

After failing to persuade their fellow Republicans last week to let taxes rise on income over $1 million, GOP leaders offered no guidance on the shape of a package the House could ultimately accept. “The House will take . . . action on whatever the Senate can pass, but the Senate first must act,” the leaders said in a joint statement.

The difference between the Democratic and Republican plans floated last week was relatively small.

The White House, meanwhile, was working with Reid on an alternative package that would keep Obama’s vow to let taxes rise on income over $250,000. Top Senate aides said their approach also would protect millions of middle-class Americans from having to pay the costly alternative minimum tax for the first time and would keep benefits flowing to 2 million unemployed workers who otherwise would be cut off in January.

The measure also could delay deep spending cuts set to strike at the Pentagon and other federal agencies next month. But aides said the scope of the package depends on negotiations with Senate Republicans, and the talks had yet to get off the ground.

With no sign of urgency, aides in both parties predicted that failure was not just a possibility — it was rapidly becoming the most likely outcome. No significant movement was expected Thursday: Obama was scheduled to be in the air traveling back from his Hawaiian holiday for a good portion of the day, and the Senate wasn’t set to convene for votes until the evening.

Unless the House and the Senate can agree on a way to avoid the “fiscal cliff,” more than $500 billion in tax increases and spending cuts will take effect next year, potentially sparking a new recession.

In a sign of mounting anxiety over Washington gridlock, Starbucks is asking employees in its D.C. area stores to write the words “Come Together” on cups provided to customers Thursday and Friday. The aim is to send a message to lawmakers that they must settle their differences and resolve the fiscal crisis, Starbucks chief executive Howard Schultz said in a letter posted online.

Financial markets, already unsettled by the prospect of dramatic tax increases and spending reductions, may also face a new battle over the limit on federal borrowing. Treasury Secretary Timothy F. Geithner announced Wednesday that the debt will hit the $16.4 trillion cap on Dec. 31, leaving roughly two months for Congress to raise it or default on the nation’s obligations.

Some lawmakers are looking ahead to the debt-ceiling fight as the moment when party leaders will finally agree on a plan to get rampant government borrowing under control.

“I think we’re going to fall out of the fiscal tree,” Sen. Lindsey O. Graham (R-S.C.) said Sunday on NBC’s “Meet the Press.” “The big chance for a big deal is at the debt ceiling. That’s when we will have leverage to turn the country around, prevent it from becoming Greece, and save Social Security and Medicare.”

Until last week, much of Washington had hoped that a broad deal would come together by New Year’s Eve. Obama and Boehner had been talking since the November election about a compromise that would meet the president’s demand for higher taxes on the wealthy and the speaker’s demand for significant changes to the health and retirement programs, such as Medicare and Social Security, that are the federal government’s biggest expenses.

The two leaders appeared to come close. In his last offer, Obama proposed to cut $400 billion from federal health programs through 2022 and to slow cost-of-living increases for Social Security benefits. All told, he proposed net spending cuts of about $850 billion in exchange for about $1.3 trillion in new revenue.

Boehner offered to raise $1 trillion in fresh revenue, and he wanted spending cuts of equal size. By that measure, Obama’s tax offer was $300 billion too high and his cuts $150 billion too low, for a net difference between the two men of about $450 billion — less than 1 percent of projected federal spending over the next decade.

In the end, however, the gap proved to be much wider politically than it was numerically. Triumphant Democrats, who gained seats in the House and the Senate in addition to holding the White House in November, were unwilling to push the tax number any lower. And they adamantly opposed additional health savings that might reduce benefits for the elderly, such as raising the eligibility age for Medicare from 65 to 67, or jeopardize the 2014 launch of Obama’s health-care initiative.

Boehner, meanwhile, might have been able to accept a higher tax target, Republicans said. But he needed parity in spending cuts, and the president did not provide it.

Democrats argue that the speaker couldn’t have sold any deal to his anti-tax caucus, pointing to the failure last week of his “Plan B,” which would have raised taxes on millionaires, after he broke off talks with Obama. Doubts about the House are now clouding the prospects for action in the Senate.

The most likely way forward, aides in both parties said, is for Reid to take up the House bill and amend it to let taxes rise for the wealthy, to extend federal benefits for the long-term unemployed and to make other urgent changes. At this late date, aides said, such a bill would be extremely unlikely to include significant spending cuts or to raise the debt limit, but it would avert the worst economic effects of the fiscal cliff.

Nearly a dozen Senate Republicans have expressed support for such an approach, including Sen. John Cornyn (Tex.), the incoming No. 2 Republican leader.

“One of the beauties of that bill is it wouldn’t require a vote to raise anybody’s taxes, and I think that is a major advantage,” Cornyn said Friday, noting that an “aye” vote would extend expiring tax cuts for most households, letting taxes rise on the rich by default.

But Senate Democrats say they would not advance such a bill until they secured McConnell’s commitment not to block it and Boehner’s pledge to let it pass the House with Democratic votes.

“House Republicans pushed middle class families closer to the cliff by wasting an entire week with their incompetent ‘Plan B’ stunt,” Reid spokesman Adam Jentleson said in a statement. “The Senate has already rejected House Republicans’ Tea Party bills, and no further legislation can move through the Senate until Republicans drop their knee-jerk obstruction.”

Wall Street traders, meanwhile, began bracing their clients for disaster.

“The blame game could reach full swing by the weekend,” the Washington Research Group, an arm of Guggenheim Partners, wrote Wednesday in its daily message to investors. “We . . . continue to believe there is a 70 percent chance that we go over some form of the cliff at the end of the year.”