Highlights from the president's speech on the government shutdown at M. Luis Construction company in Rockville, Md. Thursday. (The Washington Post)

With the government shuttered and a deadline for raising the debt limit just two weeks off, anxious Republicans began steering the party away from a dead-end debate about the health-care law and toward discussion of a broader deal to reduce the nation’s debt.

With a resolution appearing far from imminent, President Obama on Thursday night decided to cancel his trip to Asia, which was scheduled to begin Saturday.

In meetings with small groups of rank-and-file lawmakers, House Speaker John A. Boehner (R-Ohio) has emphasized that he will not permit the country to default for the first time on its debt. Given that a bloc of hard-line conservatives is unlikely to vote to increase the limit under any circumstances, Boehner has told fellow Republicans that they must craft an agreement that can attract significant Democratic support.

“This needs to be a big bipartisan deal,” Rep. Tom Cole (R-Okla.), a close Boehner ally, said as he emerged from a luncheon meeting in the speaker’s office Thursday. “This is much more about the debt ceiling and a larger budget agreement than it is about Obamacare.”

One lawmaker, speaking on the condition of anonymity, said Thursday that Boehner has even suggested that he may be willing to risk the fury of conservatives by relying on a majority of Democratic votes — and less than a majority of Republicans — to pass a debt-ceiling increase. Doing so would recall the vote tallies on the huge political defeats Boehner suffered earlier this year as he agreed to head off year-end tax increases and provide federal relief to victims of Hurricane Sandy.

See why some House Republicans might support a clean budget bill.

A senior aide denied that Boehner has suggested such a strategy. Meanwhile, senior policy aides were at work on last-ditch alternatives that could win the support of a majority of Republicans, such as increasing the $16.7 trillion debt limit for a short period — mere days or weeks — to force Democrats to the negotiating table.

“Speaker Boehner has always said that the United States will not default on its debt, but if we’re going to raise the debt limit, we need to deal with the drivers of our debt and deficits,” Boehner spokesman Michael Steel said in a written statement.

The shift in strategy among Republicans caused a brief sensation Thursday, as political analysts speculated that Boehner, who has long acknowledged the dangers of a default, may be ready to give up the fight. But Republicans pushed back hard against that idea. Although it may have been a mistake to shut down the government in a fruitless quest to dismantle Obama's health-care law, they said, that does not mean they will roll over on the debt limit without concessions from Democrats.

“I don’t think there’s energy in the Republican caucus to have any kind of default on the debt limit,” said Rep. James Lankford (R-Okla.), a member of the House leadership who represents the conservative class of 2010. But “we need a long-term plan to take future debt ceiling increases off the table.”

Increasing the debt limit for “a couple days, a week” would be a “horrible way” to jump-start talks, Lankford said. “We’d rather just get it resolved.”

A very short-term increase in the debt limit could wreak havoc on the economy and financial markets by dragging the debate well beyond Oct. 17, when the Treasury will exhaust its borrowing authority and begin relying entirely on incoming revenue to pay the nation’s bills.

On Thursday, the Treasury Department released a report on debt ceiling “brinksmanship” that examined the economic fallout from a similar impasse in 2011, when the nation came within days of defaulting on its obligations.

The report found that consumer confidence plummeted in the months before the 2011 debt limit deadline. So did the stock market, robbing the nation of $2.4 trillion in household wealth, including $800 billion in retirement assets. Interest rates also spiked, raising borrowing costs for home buyers, businesses and consumers, with the effects lingering long after the crisis was resolved.

“Playing games” with the debt limit by approving a short-term increase “would be a very, very serious mistake,” said House Minority Whip Steny H. Hoyer (D-Md.). At a time when global investors are already worried “that America has gone into a psychology where it’s prepared not to pay its bills, this would be another self-inflicted wound.”

So far, however, Obama and Senate Democrats have refused to negotiate with Republicans, either over the debt limit or a plan to fund the government into fiscal 2014, which began Tuesday. Congress reached an impasse after Republicans insisted that any funding plan include provisions to dismantle the health-care law, which began enrolling consumers Tuesday.

Since the shutdown began, many House Republicans have joined a majority of their colleagues in the Senate in publicly condemning that strategy, arguing that Obama is never going to sign a bill that undermines his most important legislative achievement. On Thursday, Rep. Charlie Dent (R-Pa.) proposed to fund the government for six months in a bill that does not mention dismantling the health-care law and seeks only to repeal a 2.3 percent tax on medical devices.

But Republican leaders have declined to endorse Dent’s bill and have refused Democratic demands to pass a simple funding bill with no other provisions, calling instead for bipartisan negotiations aimed at reopening the government and crafting a deal to raise the debt limit.

“Nobody’s doing well under this. We aren’t, but neither are they,” said House Ways and Means Committee Chairman Dave Camp (R-Mich.), who has been working with House leaders and Budget Committee Chairman Paul Ryan (R-Wis.) on a negotiating strategy for the debt ceiling. “I think they know that. And they know what’s coming.”

Frustrated Democrats say they are willing to negotiate and have been trying for months to persuade Republicans to name a conference committee to discuss federal budget issues.

Both sides agree that a potential deal could involve replacing deep budget cuts known as the sequester with cost-saving adjustments to Social Security and Medicare, such as using a less generous measure of inflation to calculate cost-of-living changes.

Both parties are willing to discuss an overhaul of the tax code, so long as the question of how to raise revenue is left open. Democrats say they would even consider changes to the health-care law, such as a repeal of the 2.3 percent tax on medical devices.

But Democrats — noting that they already have agreed to maintain the sequester cuts through mid-November — say they will not discuss a broader budget deal until Republicans agree to fund the government.

“We do need to sit down. I’m happy to talk to them,” said Senate Budget Committee Chairman Patty Murray (D-Wash.). “But the government has to open up first.”

Zachary A. Goldfarb contributed to this report.