Republican House Speaker John Boehner has vowed not to raise the US debt ceiling without a "serious conversation" about what is driving the debt.
Democrats have said it was irresponsible and reckless to raise the possibility of a US default.
Mr Boehner told ABC's "This Week," that: "The nation's credit is at risk because of the administration's refusal to sit down and have a conversation."
He added: "That there were not enough votes in the House of Representatives to pass a "clean" debt limit bill, without any conditions attached."
Mr Boehner was asked if that meant the US was headed towards a default if President Barack Obama did not negotiate ahead of the 17 October deadline to raise the debt ceiling, he said: "That's the path we're on."
The comments appeared to mark a hardening since late last week.
Mr Boehner was reported to have told Republicans privately that he would work to avoid default.
He told Republicans that even if it meant relying on the votes of Democrats, as he did in August 2011.
Republicans and Democrats also traded blame for a shutdown that has brought much of the government to a standstill for nearly a week.
With no end in sight, the battle over funding the government has merged into the one over the debt ceiling.
Republicans have demanded that Democrats agree to delay implementation of the Affordable Care Act.
The landmark 2010 law popularly known as Obamacare, as part of any spending bill.
They have also been seeking measures to address the federal government's long-term debt in exchange for raising its $16.7 trillion (€12 trillion) debt limit.
If the borrowing cap is not increased, the US could go into default.
Officials and economists say if the US goes into default it would be seriously damaging consequences for the US and global economies.
"I don't want the United States to default on its debt,"Mr Boehner said.
"But I'm not going to raise the debt limit without a serious conversation about dealing with problems that are driving the debt up. It would be irresponsible of me to do this."
Treasury Secretary Jack Lew warned of serious consequences if "the unthinkable" were to happen and the United States defaulted.
"It is irresponsible and it is reckless to take that chance, which is why Congress needs to act," he told Fox News.
He told CNN: "On the 17th we run out of our ability to borrow, and Congress is playing with fire."
Senator Ted Cruz, who has been the standard-bearer for Republican opposition to funding the government without measures to undercut Obama's healthcare law.
He told CNN that Congress frequently in the past had attached curbs on spending to votes to raise the debt ceiling.
On this occasion, he said, Republicans were looking for three things before raising the debt ceiling: a significant structural plan to reduce government spending, no new taxes, and measures to "mitigate the harm from Obamacare."
Democrats vow not to negotiate on the funding bill or the debt ceiling, arguing that it is the job of Congress to pay its bills.
Concerns that the shut down could trim economic growth coupled with nervousness over a potential debt ceiling crisis later this month have weighed on stocks and pushed the US dollar close to an eight-month low.
While selling has been orderly so far, investors expect volatility to rise as the shut down continues and the debt ceiling deadline nears.