The solution advanced by Republicans is to increase private-sector competition, by allowing health plans to sell their coverage across state lines, for example. Most experts say that's unlikely to significantly drive down America's record-high healthcare costs, however, and risks leaving in place the current problems of uninsured and underinsured patients that the healthcare reform law attempted to address in the first place.

Cantor appeared to go further than Republicans have in the past by acknowledging that not all patients are certain to get optimal healthcare under a system of private insurance.

"I think that the fundamental nature of our system of third-party payer is the problem," he said. Patients, he added, too often are left with "no decision about what they want and what they can afford."

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Later, Cantor said Republicans want a safety net for people who can't afford care but that "we're not for everyone having the same outcome guaranteed."

He also said the upcoming vote on raising the debt ceiling is "the biggest leverage moment of this [legislative] session" for dealing with entitlement reform.

"We're going to make sure that we have firm commitments and the ability to deliver on those commitments," he said.

Cantor stopped short of saying overhauling Medicare would have to be part of any deal on the debt ceiling, but he spent much of his remarks beforehand warning of out-of-control Medicare spending due to aging baby boomers and medical inflation.

He later told reporters that triggers and promises have failed in the past as lawmakers overrode them, so he wants to see "real reform, real cuts this year."