GOP Rep. Charlie Dent of Pennsylvania told CNN Chief Congressional Correspondent Dana Bash that whichever party was to blame, a shutdown will make everyone look bad.
"At this point, the hourglass is nearly empty," Dent said. "Now that we've sent over two volleys to the U.S. Senate and they rejected both."
Trying again would only yield the same result, he continued, adding that "sooner or later, we have to pass a clean resolution to fund the government before the end of the day."
"I believe the votes are there to do it," Dent said.
Obamacare a GOP focus
Obama and Democrats reject what they call Republican efforts to use the threat of a government shutdown to force negotiations on the president's signature health care reforms.
Noting that the 2010 Affordable Care Act has been upheld by the Supreme Court, they say it is settled law that voters endorsed last year by re-electing Obama over GOP candidate Mitt Romney, who campaigned on repealing it.
A new CNN/ORC poll shows that Americans are not happy about the prospect of a shutdown, which is happening because Congress has been unable to pass a budget for the new fiscal year that begins Tuesday.
According to the poll, 68 percent of Americans think shutting down the government for even a few days is a bad idea, while 27 percent think it's a good idea.
And it appears most Americans would blame congressional Republicans for a shutdown: Sixty-nine percent said they agreed with the statement that the party's elected officials were acting like "spoiled children."
Democrats, however, weren't far behind: Fifty-eight percent of respondents said they too were acting like spoiled kids.
A poll later showed public support for Congress at record low levels.
Stock traders also seemed solidly against a shutdown. The Dow Jones Industrial Average fell by more than 120 points, or nearly 1 percent, and the other two major indexes also closed lower.
Among major economic issues that could result from a shutdown: delays in processing FHA housing loan applications -- a potential drag on the housing recovery -- and the potential loss of government spending that's helping prop up the economy, said Christine Romans, host of CNN's "Your Money."
"You've got an economy right now that's very tied to government spending and government contracts, so that could have a ripple effect all across Main Street," she said on CNN's "New Day."
If the government does shut down, it would be the first time it has happened in more than 17 years. That previous shutdown, sparked by a budget battle between Democratic President Bill Clinton and a Republican Congress, lasted for 21 days.
While the military will remain on duty, as will many essential public safety, health and welfare operations, many government offices will close. About a quarter of the federal government's 3.3 million employees -- those frequently referred to as "nonessential" -- will be told to stay home from work until the shutdown is over.
Attorney General Eric Holder and Defense Secretary Chuck Hagel said essential crime prevention and military services would continue, but some workers would be furloughed. Holder said he would cut his pay by the same amount as the most severely affected Justice Department employees because "we are all in this together."