WASHINGTON—The government shutdown enters its second week with the two parties still bitterly divided and Republicans increasingly tying the fight to a fast-approaching deadline to avoid a default on U.S. debt.

The GOP got more of the blame for the government shutdown and made no progress stopping Obamacare, but the party now has an opportunity to put the focus back on spending, where it has had some success in the past. WSJ's Patrick O'Connor and Carol E. Lee explain.

House Speaker John Boehner (R., Ohio) said Sunday he wouldn't bring up bills to fully reopen the government or increase the country's borrowing limit unless Democrats agree to broader talks aimed at trimming the deficit. The speaker insisted he couldn't muster enough votes to pass either one without the concessions.

"The votes are not in the House to pass a clean debt limit, and the president is risking default by not having a conversation with us," Mr. Boehner told ABC in his first interview since the shutdown began. "I'm not going to raise the debt limit without a serious conversation about dealing with problems that are driving the debt up."

The fight to this point has centered on Republican demands to delay or dismantle parts of the 2010 health-care law in exchange for funding the government. Now, by pairing the standoffs over funding the government and raising the debt ceiling, the speaker is trying to force President Barack Obama and Senate Majority Leader Harry Reid (D., Nev.) to agree to GOP priorities on deficits and federal spending in return for movement on both. The two Democrats have repeatedly rejected GOP attempts to use the mid-October deadline to increase the debt limit as a bargaining chip.

White House officials responded to Mr. Boehner's comments in a series of Twitter messages. Deputy press secretary Josh Earnest said Republicans "don't have to give up anything. Just pay nation's bills on time, no strings attached." Mr. Reid challenged Mr. Boehner's view that he lacks the votes to pass a government-funding bill without Republican priorities.

"I think he does," Mr. Reid said via Twitter. "Let the House vote, and we'll find out."

Mr. Boehner faces two challenges from inside his party. Centrist Republicans, eager to fully reopen the government and avoid a default, have already reached across the aisle to explore possible solutions, while conservatives are threatening to turn their back on the speaker if a broader deficit-reduction deal doesn't include changes or delays to the new health-care law.

The tough talk from Mr. Boehner is as much about preserving his leverage with Democrats as it is about keeping Republicans unified. A group of House Republicans—members say there are 15 or more—have expressed frustration with the shutdown and urged party leaders to put the episode behind them. But the bloc has bypassed opportunities to vote against Mr. Boehner, raising questions about whether it would break with party leaders.

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Senate Republicans have seized on the impasse to push their own outlines for deficit reduction. Sen. Rob Portman of Ohio is trying to enlist support from Senate Democrats for a proposal he believes could break the stalemate, according to a person familiar with the plan. Mr. Portman has recommended an agreement that would fund the government for one year at the level favored by House Republicans that would include plans to start lawmakers on an overhaul of the tax code. His plan also would raise the debt limit by an amount equal to deficit-reduction measures already backed by Mr. Obama in his most recent budget, including a switch to a new measure of inflation under which Social Security benefits would rise more slowly.

Sen. Chuck Schumer (D., N.Y.) said in an interview that five Republican senators had approached him about trying to broker an end to the shutdown. He declined to elaborate, saying only, "They know that it's bad for both their party and country to keep the government shut down, and they know it's even worse for their party and their country to default."

Mr. Schumer and other Democrats saw hints of flexibility in the speaker's remarks Sunday.

"I will predict to you today that Boehner will not let us default," said Mr. Schumer, a member of the Democratic leadership team, pointing to the speaker's request to have a "conversation" as evidence of his willingness to strike a deal on Republican priorities. "Default is so calamitous that even if there's only a 30% to 40% chance of it, it freezes the markets and pulls us into recession. Boehner can't risk this, and he knows it."

Some conservatives view the prospects of broader budget talks with suspicion, considering it a dodge for GOP leaders to back away from the fight to undercut the health law. Mike Needham, head of the conservative political group Heritage Action, said talk of a broader deficit deal is "Washington-speak" for undermining the next round of automatic spending cuts known as the sequester, increasing taxes and adding more debt.

"Congress should ensure not another dime of taxpayer money is spent on Obamacare," Mr. Needham said.

Other Republicans said the time had come for conservatives to relent in trying to link government funding with measures to dismantle the health law. Senate Democrats have rejected multiple bills approved by House Republicans to fund federal agencies while also delaying or scaling back the health law.

"The question now is whether the people who fell for the Cruz folly will recognize that it was built on a false premise," said Rep. Devin Nunes (R., Calif.), referring to Texas Sen. Ted Cruz and other conservatives who pushed Republicans to call for defunding the health law as a condition of financing government operations. "The whole thing was a joke from the beginning."

There is no guarantee that a broader budget deal could be crafted even if Republicans shelved their demand to amend the health law. Sen. Bob Corker (R., Tenn.), who took part in private budget talks with the White House over the summer, said those discussions didn't inspire confidence in future negotiations. "I hate to say this: Instead of building trust, they probably broke trust down even further," he said.

The Tennessee senator said he expects the president to engage with GOP leaders now that House Republicans are unifying around a common position insisting on a broader discussion about the nation's debt. "He doesn't have to negotiate as long as [House Republicans] are in disarray," Mr. Corker said in an interview. "If they can unify around some sensible, pragmatic fiscal goals, then the president has to—has to—negotiate."

The impact of the shutdown is still rippling through the broader economy. The Pentagon said over the weekend it would recall most of the 350,000 civilian employees who had been furloughed as a result of the shutdown, a big chunk of the roughly 800,000 federal workers who are temporarily out of work.

But the effects are being felt further. Housing-industry officials, for example, predict a lengthy shutdown could make it tougher for home buyers to secure mortgages, in part because of reduced staffing at the Federal Housing Administration.

It remains unclear at this point what Mr. Boehner and other GOP leaders want to see emerge from any broader budget talks. Sunday, the speaker ruled out additional tax revenue, which Mr. Obama has said would have to be paired with any effort to scale back entitlements.

"He got his revenues," Mr. Boehner said on ABC, referring to the deal Congress approved at the beginning of the year to roll back tax cuts for people making more than $400,000. "We're not raising taxes."

In the interview, the speaker also acknowledged publicly that the fight over funding the government wasn't his choice but rather that of his rank-and-file, an unusual confession for such a high-profile politician. "I thought the fight would be over the debt ceiling," he said. "Working with my members, they decided, well, let's do it now. And the fact is, this fight was going to come, one way or the other."

Asked when he thought the standoff would end, Mr. Boehner offered only uncertainty, saying, "If I knew, I would tell you."

—Damian Paletta and Janet Hook contributed to this article.

Write to Peter Nicholas at peter.nicholas@wsj.com