A clean debt-limit extension is not yet on the GOP’s radar, though Boehner has said he would avoid default.

Other proposals, such as requesting a balanced-budget amendment in exchange for an extension, still have broad support in Republican ranks, but have been momentarily shelved, since they are unlikely to win even scattered Democratic support.

In an interview, Rep. Tom Cole (R-Okla.) said Boehner would spend this week “looking for the sweet spot” as his colleagues mull how to proceed. “After the State Department’s report on the pipeline, that becomes an easier pitch,” he said. “But the risk corridors item was popular at the retreat and remains popular.”

Risk corridors, a provision of the ACA, limits 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. Related programs that mitigate risk for insurance companies are also being targeted by conservative Republicans.

“We’re in the process of sorting it out,” Cole said. “The number one question is: how do we get to 218 votes? We want to do this in a non-crisis atmosphere, so we have to move soon and work with the administration and see if they want to do anything.”

Growing tension, however, is already apparent between the two strategies and their supporters. Some conservatives have said they would urge Boehner to link repeal of the risk corridors to any debt-limit demand, believing the federal health-care law must be part of the negotiations. Other Republicans have said tying the pipeline to the debt limit is a better play in an election year.

Aides said Tuesday morning’s open-microphone session near the end of the closed-door meeting would be a good indicator of which bargaining chip is winning the most converts.

In the meantime, members are buzzing about the two options, as well as grappling with the reality of divided government and the desire to keep the often-fractious House GOP united.

Rep. Steve Scalise (R-La.), the chairman of the Republican Study Committee, said merely connecting the pipeline to the debt limit would not guarantee conservative backing.

“Keystone ought to be approved on its own merits, it doesn’t need to be tied to the debt ceiling,” he said. “I don’t think that would be enough for a lot of RSC members to support.”

“I’d like to see a repeal of the bailouts for the insurance companies,” he added, referencing the risk corridors. “That’s becoming a real strong prospect; it’d be good to put it on the table.”

House GOP aides said they expect the conference to reach a consensus by later this week, and said they would not be surprised if the leadership softly whips both options to see which is the best way to keep Republicans moving together.