Updated 12:59 p.m.
The proposal signals that political fights in the coming weeks are unlikely to focus on whether to raise the nation's borrowing limit — just on whether congressional Republicans will demand concessions from the White House to do so.
Under one scenario discussed Friday morning at the House GOP's annual policy retreat held on Maryland's Eastern shore, Republicans would agree to extend the debt limit for one year, but demand that there be "no bailouts for insurance companies under Obamacare," the lawmaker and aides said. House Majority Leader Eric I. Cantor (R-Va.) and House Budget Committee Chairman Paul Ryan (R-Wis.) described to colleagues how this scenario could play out and conservative lawmakers in the room seemed supportive of the idea, including Rep. Michele Bachmann (R-Minn.), who spoke up in support and offered to help whip up support for the plan among Republicans.
But the aides stressed that Republican leaders have "not decided" or formalized a strategy and that no formal consensus would emerge Friday.
Concerns about a so-called "Obamacare bailout" have emerged in recent days, especially on conservative op-ed pages. The term is generally used to describe three programs in the health-care law — two temporary and one permanent — that make it less financially risky for health insurance plans to sell on the new exchanges. The term also is sometimes used to refer to one specific program in the health-care law known as "risk corridors" that limit both the amount of money that a health insurance plan can make and lose during the first three years it is sold on the new health-care exchanges established by the law.
More generally, risk corridors are a pretty common feature of insurance design. They're still used in Medicare Part D, the program where private insurers sell prescription drug plans to seniors.
Even if Republicans settle on a strategy, it is expected to be met with resistance by President Obama and congressional Democrats, who continue to insist that the debt ceiling must be increased with no stipulations.
To reiterate the strategy and ensure unity in the coming weeks, Treasury Secretary Jacob J. Lew and White House communications director Jennifer Palmieri reminded Democratic senators Thursday that the White House is "not paying ransom" to congressional Republicans to secure another increase in the federal debt limit. Lew and White House Communications Director Jennifer Palmieri reiterated the administration's position during a weekly Democratic policy luncheon.
Senators emerged from the meeting in agreement with the White House.
"There is no reason to hold anything hostage to getting a debt ceiling raise," Senate Budget Committee Chairman Patty Murray (D-Wash.) told reporters. "We just had a budget agreement, we had an appropriations bill, we determined where our spending is going to be, we now have a responsibility as the managers of this country to pay for that."
Another option mentioned at the closed GOP meeting Friday included a demand for a constitutional amendment to balance the federal budget in exchange for a debt-limit extension, though even conservatives openly conceded to their colleagues that with Democrats controlling the White House and Senate, such a demand is unlikely to be viable. "Some other old conservative chestnuts entered the conversation but there was not a rally to those demands," said a second House Republican, requesting anonymity to discuss confidential discussions.
Paul Kane and Sarah Kliff contributed to this report.