Conservative Republicans in the House and Senate are demanding that any government funding bill include language defunding ObamaCare and have said they will oppose a funding measure that doesn't include the ObamaCare defunding language.
Since Senate Democrats and President Obama are not expected to accept such language, the bill would likely lead to a government shutdown Oct. 1.
On Wednesday, GOP leaders in the House delayed a measure that aimed to placate members seeking to defund the healthcare law. GOP leaders said more time is needed to build support for the proposal.
Many conservatives have rejected the plan, which would force the Senate to vote on defunding ObamaCare, but would not require that proposal to pass in order for the overall funding bill to be sent to the president.
If the government were to shutdown for a few days, 49 percent of people would see it as either a major problem or a crisis, according to the CNN poll. Forty percent would see it as a minor problem.
But if the government remained closed for weeks, 74 percent would call it a major problem or crisis, while those describing it as a minor problem drops to 20 percent, the poll found.
The healthcare law remains deeply unpopular, according to the poll.
Only 6 percent of people say they favor all parts of the law. Thirty-three percent say they favor most of it, while 57 percent say they oppose all or most of it.
The poll surveyed 1,022 national adults and contains a margin of error of 3 percent.