About 56 percent of respondents said it would be bad for the country if the debt limit was not raised, the CNN/ORC International poll found. Just 38 percent said failure to increase the limit would be a good thing.
Asked who would be responsible if the limit were not raised, 53 percent said congressional Republicans and 31 percent said President Barack Obama.
Raising the debt limit looms as the next Washington battle as Republicans continue to fight with Democrats and the White House over passing a spending bill that would end the first partial federal government shutdown in 17 years.
The poll results came as Treasury Secretary Jacob J. Lew wrote to congressional leaders Tuesday night that his department had just about exhausted all the so-called extraordinary measures it has been using since May, when the nation technically hit its borrowing limit.
The measures, including suspending investments in some federal pension funds and in a currency exchange rate fund, allowed the Treasury Department to juggle the nation's finances and continue to pay bills.
But Lew wrote that "there are no other legal and prudent options to extend the nation's borrowing authority," and he reiterated his recent warning that Congress needed to act by Oct. 17 or risk the country's first-ever default.
"It is important to note that once the final extraordinary measures are exhausted, no later than October 17, we will be left to meet our country's commitments at that time with only approximately $30 billion," he wrote of the amount of money the Treasury would have on hand to pay bills of as much as $60 billion on any given day.
"If we have insufficient cash on hand, it would be impossible for the United States of America to meet all of its obligations for the first time in our history," Lew wrote. "For this reason, I respectfully urge Congress to act immediately to meet its responsibility by extending the nation's borrowing authority."
Republican lawmakers have said they won't raise the debt limit without concessions from Obama. House Speaker John Boehner, R-Ohio, wants spending cuts or reforms at least equal to the amount of the debt limit increase.
Some conservatives have said they want to eliminate funding for Obama's health care law or delay its implementation - the same demands they have made for passing a government spending bill.
The CNN poll found that by a 51-to-43 percent margin, respondents said it was more important to raise the debt limit than delay the health care law.
Chris Krueger, a senior policy analyst at financial services firm Guggenheim Partners in Washington, said Wednesday that he believed there was a 40 percent chance Congress would not pass a debt limit increase by Oct. 17.
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