Both sides now concede that tax revenue and reductions in entitlement spending are essential elements of any deal. If the talks succeed, it probably will be because House Speaker John Boehner yields on raising tax rates for top earners and the White House bends on how to reduce spending on Medicare and accepts some changes in Social Security.

The White House and Boehner kept up the ridicule of each other’s negotiating stances on Tuesday. But beneath the tough words were the possible makings of a deal that could borrow heavily from a near-bargain last year during debt-limit negotiations.

Then, Obama was willing to reduce cost-of-living increases for Social Security beneficiaries and increase the eligibility age for Medicare, as Boehner and other top Republicans have demanded. On Tuesday, Obama did not shut the door on Republican ideas on such entitlement programs.

“I’m prepared to make some tough decisions on some of these issues,” Obama said, “but I can’t ask folks who are, you know, middle class seniors who are on Medicare, young people who are trying to get student loans to go to college, I can’t ask them to sacrifice and not ask anything of higher income folks.”

“I’m happy to entertain other ideas that the Republicans may present,” he added in an interview with Bloomberg Television.

At the core, the negotiations center on three key points: whether tax rates for upper income taxpayers should go up, how deeply to cut spending on entitlements such as Medicare and how to deal with raising the government’s borrowing limit early next year.

White House spokesman Jay Carney dismissed Boehner’s proposals as “magic beans and fairy dust.”

Boehner countered: “If the president really wants to avoid sending the economy over the fiscal cliff, he has done nothing to demonstrate it.”

Tax rates have emerged as one of the most intractable issues, with Obama insisting the rates on the top 2 percent of earners must go up and Boehner standing steadfast that they must not.

Boehner, instead, has proposed raising $800 billion through unspecified loophole closings and limits on tax deductions.

On Tuesday, the president said he would consider lowering rates for the top 2 percent of earners — next year, not now — as part of a broader tax overhaul effort that would close loopholes, limit deductions and find other sources of government revenue. “It’s possible that we may be able to lower rates by broadening the base at that point,” Obama said.

On Medicare and Social Security, the Republican proposals would do relatively little to curb the deficit over the next decade, but the impact would grow over the longer term.

Raising the Medicare retirement age from 65 to 67, for instance, would wring $148 billion from the program over 10 years, according to a Congressional Budget Office estimate last year, about one-fourth of the savings House Republicans hope to claim from federal health programs.

Another idea that gained currency during the Obama-Boehner talks last year would change the annual inflation measure used for Social Security cost-of-living increases and the indexation of tax brackets for inflation.

Many economists and government budget specialists believe the system is a more accurate measure of inflation because it takes into account changes in purchasing behavior

This “chained consumer price index” idea makes modest cuts to Social Security benefits at first — curbing program costs by $112 billion over a decade according to the 2011 CBO report. But those reductions build up more over time in a fashion comparable to the way compound interest builds personal savings.

The White House has not foreclosed the idea of addressing Social Security cost-of living changes in a new deal, but it has not embraced it because Obama’s aides argue Social Security is not contributing to the federal deficit.

The stingier inflation measure also could raise tax revenue by $87 billion over the coming decade. Taxes would slowly increase because annual adjustments to income tax brackets would be smaller, pushing more people into higher brackets.

But the alternative inflation measure, while a favorite of budget hawks, has run into fierce opposition from defenders of Social Security.

“I’ve never been a part of that,” said Senate Majority Leader Harry Reid, D-Nev., a top Obama ally.

The two sides are also close, at least in theory, on curbing spending on a host of miscellaneous programs, as well as new fees. These could lead to higher airline ticket prices, for example, an end to Saturday mail delivery, fewer food stamps and lower farm subsidies.

Republicans claim they could glean $300 billion from such cuts and fees over 10 years; the White House promises $250 billion.

So far, the public seems ready to hold Republicans responsible if negotiations fail. A new Washington Post-Pew Research Center poll shows that 53 percent say the Republicans would deserve blame if the nation tips over the fiscal cliff, and only 27 percent of those surveyed say Obama would be to blame.

Forty-nine percent don’t believe Obama and Congress will reach a deal by Jan. 1, whereas 40 percent are more optimistic.

Republicans were quick to say on Tuesday that Boehner’s plan was attracting criticism from the right, particularly from Republican Sen. Jim DeMint of South Carolina, a leader of tea party conservatives, and as such represented more of a compromise than Obama’s stance. DeMint said Boehner’s plan “will destroy American jobs and allow politicians in Washington to spend even more.”