House Speaker John Boehner (R-Ohio), Rep. Lynn Jenkins (R-Kansas) and Rep. Cathy McMorris Rodgers (R-Wash.), arrive to speak to reporters about the Keystone XL Pipeline and other issues following a Republican conference on Tuesday in Washington. (J. Scott Applewhite/AP)

House Republicans were still struggling Tuesday to find consensus on how to handle their upcoming debt-limit negotiations with the White House, but they seem increasingly determined to avoid any kind of dramatic showdown with the president this time.

“We should bring up a clean debt ceiling, let the Democrats pass it and just move on,” said Rep. Raúl R. Labrador (R-Idaho). “Our constituents are fed up with the political theater. If we’re not going to fight for something specific, we might as well let the Democrats own it.”

The House leadership spent much of the day debating how to leverage their power to extract concessions from the president in exchange for raising the federal government’s borrowing authority.

“We talked to members last week about it, talked to them again today about it, “ House Speaker John A. Boehner (R-Ohio) told reporters at a news conference. “No decisions have been made.”

Much of the caucus, however, seemed to be coalescing around the idea of linking a one-year extension of the debt ceiling to another effort to repeal some provisions of the Affordable Care Act.

But there was strong support — even among some of the most conservative members — for avoiding a dramatic partisan standoff on the debt limit, signaling a break from the combative fiscal politics of the recent past.

The bargaining idea that appears to have the most momentum is a repeal of the provision in the Affordable Care Act known as “risk corridors,” which limit 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. Republicans see the corridors as bailouts for the insurance industry if the plans start losing money.

A second option being considered by the Republican leadership would be to ask President Obama to approve the Keystone XL pipeline in exchange for a debt-limit extension.

The health-care and Keystone ideas were first discussed last week at the House GOP’s annual retreat in Cambridge, Md., a closed-door affair where House Republicans conferred about their legislative agenda.

Members have mentioned other debt-limit proposals, such as changes to the federal budgeting process, although those options are, for now, below the radar. Instead, House Republicans are focused on whether to use energy policy or health care as a bargaining chip.

In a small huddle after the conference-wide session, Boehner and other House GOP leaders privately agreed that the risk-

corridors proposal was gathering momentum. Boehner and Majority Leader Eric Cantor (R-Va.) sense that House Republicans will end up rallying behind that plan in the coming days.

Still, a “clean” debt-limit hike — one with no strings attached — has not been ruled out, and Boehner reiterated his desire to avoid a federal default.

Rep. Justin Amash (R-Mich.) agreed. “It’s theater,” he said, commenting on the latest flurry of stories about possible GOP plans. “It’s going to end up being clean anyway. I don’t see anything they can put on the table that I would support as some sort of trade-off.”

But other House Republicans are likely to first try to extract a political price from Obama and Democrats in exchange for Republican backing of an extension. And Sen. Ted Cruz (R-Tex.), who coordinates closely with many of them, urged them Tuesday to keep trying.

“Of course the debt ceiling should be used to enact structural spending reforms to fix the problem, to fix the out-of-control spending and debt. That’s what all of our constituents expect, and it’s what every member of this body tells our constituents we will do,” he said, adding later: “History makes clear that the debt limit is the only effective, or one of the few effective, lever points for meaningful spending restraint.”

Settling on a strategy that can win 200-plus GOP votes is Boehner’s chief goal, according to his aides. His whip team will fan out through the House this week to get a better sense of whether members are coalescing around a particular option. Months after the poll-sinking government shutdown and months before the midterm elections, Boehner is looking to keep the Republican drama to a minimum.

“Listen, I don’t do the vote counting,” Boehner told reporters when asked about House GOP dynamics. “The goal here is to increase the debt ceiling. Nobody wants to default on our debt. But while we’re doing this, we’ve got to do something” on jobs or the economy.

Boehner grew angry when a reporter suggested that Republicans might be pushing “for something as complex as the Keystone pipeline.”

“Complex? You think the Keystone pipeline is complex?” Boehner said, raising his voice. “It’s been under study for five years. We build pipelines everywhere in America every day. Do you realize there are 200,000 miles of pipelines in the United States? And the only reason that the president’s involved in the Keystone pipeline decision is that it crosses an international boundary. Listen — there’s nothing complex about the Keystone pipeline. It’s time to build it.”

Rep. Tom Cole (R-Okla.), a top Boehner ally, said the speaker would spend this week “looking for the sweet spot.”

“After the State Department’s report on the pipeline, that becomes an easier pitch,” Cole said. “But the risk-corridors item was popular at the retreat and remains popular.”

The growing tension is apparent. Some conservatives have said that 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 that tying the pipeline to the debt limit is a better play.

Rep. Steve Scalise (La.), chairman of the Republican Study Committee, a conservative caucus, said that 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’d like to see a repeal of the bailouts for the insurance companies. That’s becoming a real strong prospect. It’d be good to put it on the table.”

Rep. Dave Camp (R-Mich.), chairman of the Ways and Means Committee, said that Scalise’s perspective has currency in a conference that has a conservative bent.

“Chairman Scalise and the RSC are very interested in that one,” he said as he left the meeting. “It’s going to take some time. There was a lot of talk about risk corridors at the retreat, but there are some other issues we’re going to look at, too.”

Added Rep. Paul Ryan (R-Wis.), the Budget Committee chairman: “I want to get us to consensus, and we’re working on doing that.”

Boehner also said during Tuesday’s conference meeting that he would like to see the House act before the Senate, in order for the House to be in a better position to shape the final negotiated product. If the House takes too long to reach debt-limit consensus, he warned that Senate Democrats could send over their own debt-limit plan.

“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.”

Clarification: This story has been updated to clarify Boehner’s comments on discussions about the debt limit with his fellow House Republicans.