House Speaker John A. Boehner is promising a “whale of a fight” this fall. But a fight over what? That is the $16.7 trillion question as Congress barrels toward another showdown over the federal debt limit.

House Republicans are trying to figure out how to finesse the deadlines looming after lawmakers return to Washington on Sept. 9. Unless Congress acts, the federal government will shut down on Oct. 1 and, according to the Obama administration, the nation will run out of cash to pay its bills in mid-October.

Boehner (R-Ohio) has proposed a short-term budget bill to keep the government open into the new fiscal year with relatively little fuss. But during a speech in Boise, Idaho, on Monday, he said House Republicans will draw a line in the sand over lifting the federal debt limit, demanding spending “cuts and reforms that are greater than the increase in the debt limit.”

“I wish I could tell you it was going to be pretty and polite and it would all be finished a month before we’d ever get to the debt ceiling. Sorry — it just doesn’t work that way,” Boehner said, predicting a repeat of the debt-limit fight of 2011, which tanked consumer confidence, along with the GOP’s approval ratings.

“What I’m trying to do here,” he said in remarks reported by the Idaho Statesman newspaper, “is to leverage the political process to produce more change than what it would produce if left to its own devices.”

What kind of change? Senior Republican aides say it is becoming clear that Boehner will have to launch a concerted assault on the Affordable Care Act, President Obama’s signature health-insurance initiative. The Heritage Foundation, the Club for Growth and other conservative groups are demanding a full-on attempt to defund the law, and at least 80 House Republicans have signed on.

GOP leaders are resisting. Instead, talks have focused on options such as delaying the requirement that individuals purchase health insurance, which is set to take effect in January; repealing a new tax on medical devices that helps fund the law; and codifying Obama’s decision to delay penalties on employers who fail to offer insurance to workers.

This month, James Capretta and Yuval Levin, scholars at the conservative Ethics and Public Policy Center, argued in the Weekly Standard that the “individual mandate” is “the most essential part of Obamacare” and “also among the most unpopular.” This, they wrote, “is where efforts to use the GOP’s limited leverage should be concentrated.”

This week, the Heritage Foundation fired back with a blog post declaring that “delay is not enough.” But a growing cadre of senior Republicans has condemned the defund strategy as doomed to failure, and GOP insiders say Boehner is unlikely to take a stand that would essentially require Obama to nullify his most significant legislative achievement.

Other ideas are on the table but appear less likely to be included in Boehner’s opening bid on the debt limit. In Idaho, he played down the role of tax reform, and he appears not to have mentioned the Keystone XL pipeline project. White House approval of the pipeline is another potential GOP demand.

Instead, Boehner identified Obamacare as the primary target. After three straight years of trimming agency spending, which accounts for only about a third of the federal budget, Boehner said, “it’s time to deal with the mandatory side.”

So far, however, House leaders have shown little interest in tackling popular mandatory-spending programs such as Social Security and Medicare, two of the biggest drivers of federal spending and where Obama has offered to find significant savings in exchange for an agreement to raise taxes on the rich. House Ways and Means Committee Chairman Dave Camp (R-Mich.) has drafted Obama’s proposals for entitlement changes into formal legislation, but the conservative clamor for cutting Social Security and Medicare has all but evaporated.

That leaves Obamacare. The good news for Republicans is that eliminating the individual mandate would indeed reduce the deficit. Insurance premiums would be higher, and 16 million fewer people would have insurance by 2021, according to the most recent estimates by the nonpartisan Congressional Budget Office. But the move would also mean fewer people would get Medicaid or federal health subsidies that reduce their tax bills, saving the government about $300 billion over the next decade.

That is a good chunk of the roughly $700 billion Boehner will need to raise the debt ceiling through the 2014 elections under current law. An additional $1 trillion or so would get Washington through the 2016 presidential election, barring a big new expense, such as a war in the Middle East or the end of the automatic budget cuts known as the sequester.

Of course, a one-year delay in imposing the mandate would save a fraction of that amount. And no mathematical calculation would help with the politics of such a holdup. Senate Majority Leader Harry M. Reid (D-Nev.) has called the idea “insane.” And White House budget director Sylvia Mathews Burwell last week told Bloomberg TV that the administration is “not interested at all in delaying what we believe is bringing people onto health care and continuing a path of reducing [health] costs.”

Where does that leave things? Probably with the Senate, where a group of Republicans known as “the Diners Club” is scheduled to meet this week with White House officials. But those talks have been going nowhere, according to people on both sides of the table, and senior GOP aides say Reid and Senate Democrats might have to pass a debt-limit plan on their own.

Theoretically, that could throw the whole mess into a House-Senate conference committee for resolution. But with no clear path through the thicket, some Hill aides are grimly bracing for a series of bills to buy time by raising or suspending the debt limit for brief periods throughout the fall.

Democrats are accusing the GOP of fomenting another unnecessary political crisis.

“The last thing families and businesses across America want right now is another round of debt limit brinksmanship that would rattle the markets and threaten our fragile economic recovery,” Senate Budget Committee Chairman Patty Murray (D-Wash.) said in a statement.

It would also open up running room for ideas such as the Debt Ceiling Alternative Act, the latest brainchild of Rep. David Schweikert (R-Ariz.), a member of the conservative class of 2010 and a perennial thorn in the side of GOP leaders.

Schweikert’s measure would not lift the debt limit. It would exempt interest on the debt, permitting the Treasury to continue borrowing to make those payments. And it would authorize a massive sell-off of U.S. assets to pay the rest of the nation’s bills, starting with mortgages held by Fannie Mae and Freddie Mac.

The idea might be unorthodox, Schweikert said in an interview, but “what we’ve been doing, over and over, hasn’t turned out to be particularly brilliant.”

David Fahrenthold contributed to this report.