After last fall's government shutdown battle and two subsequent bipartisan fiscal deals, Republicans are focusing on demands that they believe Democrats might support. For example, a proposal to get rid of a tax on medical devices has garnered Democratic support in the past, though attaching it to the debt limit legislation would make it more controversial.
Representative Tom Price, a conservative Republican from Georgia, said he would also like to see some other adjustments to Obamacare considered as part of the debt limit debate, including to the so-called "risk corridor" provision that compensates insurers if they wind up with an especially unhealthy and costly mix of customers under the program.
(Read more: Lew warns Congress of February debt ceiling deadline)
"Nobody is interested, on our side of the aisle at least, in bailing out insurance companies. I would hope that the president isn't interested in bailing out insurance companies, and (Senate Majority Leader) Harry Reid isn't, so maybe there's some common ground there," said Price, who is a physician.
Listening to members
House Republicans will formulate their conditions for an increase in U.S. borrowing authority next week at a retreat in Cambridge, Maryland, a Chesapeake Bay resort town.
House Speaker John Boehner has recently softened his tone, calling last week for swift action by the House and Senate to raise the debt limit, and saying that the United States "shouldn't even get close to" default.
"No one wants another market-rattling showdown, but, at the same time, a 'clean' increase can't pass the House, or—most likely—the Senate," said a House Republican leadership aide who requested anonymity.
(Read more: CNBC Explains: The debt ceiling)
What House Republicans aren't talking about thus far are demands for major cuts to the federal benefits programs known as entitlements - Social Security, Medicare and Medicaid - which are widely viewed as the biggest contributors to future debt growth.
Obama and Democrats have successfully resisted changes to these programs through three years of fiscal showdowns, and Republicans appear to be changing tactics.
Still, it's unclear whether Democrats will accept even scaled-back Republican demands on the debt limit.
A senior Senate Democratic aide said Congress should simply increase the debt limit so that the United States can pay its bills on time.
"There are no talks whatsoever about considering any ransom demands from Republicans. Republicans are having this conversation entirely with themselves," the aide said, speaking on condition of anonymity.
On Wednesday, Treasury Secretary Jack Lew sought to raise pressure on Congress for swift action, saying the government would exhaust its borrowing capacity by late February.
As part of the deal that ended the shutdown last October, Congress suspended the debt ceiling until Feb.7. The Treasury can employ extraordinary cash-management measures to stave off default, but these won't last long, because February is traditionally a big deficit month as tax refunds are paid out.