WASHINGTON – When President Obama and Congress return to Washington later this week, the countdown to the fiscal cliff will be measured in days – yet no one really knows how, when or even whether an agreement might be reached.

Sorting out the scenarios is like trying to assemble a 100-piece puzzle on deadline. About the only decent bet is the one offered by Sen. Joe Lieberman, a Connecticut independent:

“We’re going to spend New Year’s Eve here, I believe,” he told CNN on Sunday.

There are four general ways the drama is likely to unfold: no deal at all, the long-elusive big deal, consideration of legislation that one chamber already passed or a small time-buying accord that would require further negotiations.

If no alternative is adopted, Bush-era income tax cuts will expire at the end of the year. On Jan. 2, $109 billion in automatic spending cuts would take effect, and half of them would involve defense.

Those are just the big items, however.

Extended unemployment benefits for an estimated 2.1 million Americans begin expiring at the end of this month. Also ending would be the 2 percentage-point Social Security payroll-tax cut and the alternative minimum tax “patch.” The AMT, originally aimed at wealthy taxpayers who avoided high taxes with deductions, wasn’t indexed to inflation, and therefore it could affect 31 million people unless Congress approves a change.

Also looming is the annual battle over the “doc fix,” or Medicare payments to doctors. If no action is taken, those payments might be cut by about 27 percent.

It’s difficult to gauge sentiment for agreements. Although Obama pleaded with congressional leaders Friday to work something out and they plan to try, Republicans in the House of Representatives remained defiant.

Even as the president used his weekly Saturday radio address to offer a happy holidays greeting, House Speaker John Boehner used much of the Republicans’ own weekly address to talk politics.

Obama, Boehner said, “refuses to challenge the members of his party to deal honestly with entitlement reform and the big issues facing our nation. That is why we find ourselves here today.”

But if the negotiating chaos of recent years has taught anything, it’s not to pay too much attention to the public sniping. Obama, Boehner, Senate Majority Leader Harry Reid, D-Nev., Senate Republican leader Mitch McConnell, R-Ky., and their staffs are expected to talk privately.

Here are the likely potential scenarios:

• No deal. Few on Capitol Hill will say it publicly, but there’s a lot of talk about how each side can benefit politically if the stalemate continues past Jan. 1. And if the markets don’t react once the cliff is reached, there might be even less incentive to get a quick deal.

There also might be a new incentive to act. Since higher tax rates would go into effect, getting a tax deal might become easier. Lawmakers could argue that they’re now cutting taxes for lower- and middle-income people while keeping them intact for the wealthy.

• Big deal. Obama and Boehner weren’t that far apart when they traded offers last week, and for all the sniping there are messages of hope in the remarks from both sides. Boehner offered $1 trillion in new revenue and the same amount in spending cuts. Obama offered $1.2 trillion in revenue and about the same in cuts. Republicans contest the president’s spending-reduction figure, however, saying that once interest savings are subtracted, the cuts are less than $1 trillion.

• Pending legislation. There are vehicles available to lawmakers that would enable them to move quickly. The House passed a measure in August to continue all the Bush income-tax rates for a year. The Senate adopted a measure in July to maintain those rates for all but individuals who earn more than $200,000 a year and families that make more than $250,000, the same limits Obama backed.

Those bills, which also contain an AMT patch, might be taken up again. Such legislation is likely to draw strong support.

• Small deal. There are more “fiscal crises” on the horizon: A new debt-limit ceiling is likely to be needed in mid-February and current fiscal-year spending runs out March 27.

Congress and the president might extend the tax rates or avoid some spending cuts and couple that with one of the other upcoming crises in the hope that a big deal could be worked out later. Such a smaller deal also would be easier to get through Congress before Jan. 1.

There’s a precedent. A year ago, conservative Republicans staged a revolt against a plan to extend the Social Security payroll-tax cut and some other expiring benefits. Certain jobless benefits were about to run out and Medicare payments were to balloon. Two days before Christmas, the House and Senate agreed to a two-month extension.