In part, that's a reference to one fundamental rift: He has demanded that any entitlement overhaul be accompanied by new tax revenue, and Republicans have said "no" with equal firmness.
For now, the big reform idea that's been pitched in public is called "premium support," championed by Rep. Paul Ryan (R) of Wisconsin.
It's a free-market approach to reform. Enrollees who reach Medicare age in 2024, or later, would receive a government "premium-support payment" that they could use to buy health coverage. Insurers would compete for their business, with benefit packages that would have to be equivalent to today's Medicare, or better.
The annual growth of the government aid – Democrats deride it as a voucher – would be capped using a process of competitive bidding by insurers.
"The benchmark … would be either the second-least-expensive private plan" or traditional Medicare, whichever costs less, according to Congressman Ryan's budget blueprint. And the support level, per person, couldn't rise faster in a year than the rate of gross domestic product growth plus 0.5 percent.
Seniors could buy higher-cost insurance if they wanted, covering the difference on their own. Or they could buy a lower-cost plan, and pocket the resulting savings.
The "GDP plus 0.5 percent" formula would be a slower growth rate than Medicare has seen in the past, but some health-care experts say it's not unthinkable to use that target and still ensure strong health benefits.
And others note this, too: Ryan pitched his plan as an "invitation" to bargain, not as a take-it-or-leave-it offer.
The idea "is fairly flexible," and has been embraced by leaders of both parties in the past, says Alain Enthoven of Stanford University, long a leading proponent of the concept. (He pushed such a reform as a consultant to the Carter administration.)
One alternative to Ryan's formula, for instance, would be for premium support to rise based on average health costs in the economy.
But others are skeptical that it does much to restrain costs. In markets in which insurers feel compelled to offer services from the same couple of hospital groups, for example, insurance prices might still remain high.
Mr. Enthoven agrees this is a risk, but still predicts that premium support would serve as a vital lever to push health companies toward lower-cost, and more-effective, delivery of care.
What's the alternative?
A budget plan released by Senate Democrats is largely silent on ways to make health-care entitlements sustainable for the long run.
But Medicare reform could be done in a variety of ways other than premium support. Mr. Obama has pronounced himself ready "to do some tough stuff," but not to "gut Medicare or gut Social Security or gut Medicaid."
A recent New York Times report said the president has talked with congressional Republicans about specific potential cost-savers, including the idea of combining the coverage of hospital costs (Medicare "Part A") and physician costs ("Part B") so enrollees have a single deductible. A hike in out-of-pocket costs would save money for the government, while also paying for a new benefit – a cap on the total out-of-pocket costs that a beneficiary might face.
A range of other reforms is possible, with some Democrats pushing to allow Medicare to bargain with pharmaceutical companies for lower drug prices. Whatever ideas are pursued, the rift over tax revenue remains a stumbling block.
Many health-care experts agree with Democrats that entitlement reform should ultimately involve increased tax revenues as well as cost control.