Kaiser's report on the poll said there is a split among Republicans "on whether there should be a Republican-sponsored replacement for the law or not."
About 3 in 10 Republicans said they want Obamacare repealed and replaced with a Republican alternative, the survey found. But 25 percent of Republicans say they don't want the law replaced after it is repealed.
Hamel said the survey was conducted before House Speaker Paul Ryan, R-Wis., announced his proposals for a replacement to Obamacare last week.
Ryan's proposals are likely to find more support from Republicans than with a broad cross section of Americans. Kaiser's survey found that just 33 percent of all respondents want Obamacare repealed, while 28 percent want to expand what the law does.
Only 11 percent "want to scale back what the law does," according to Kaiser.
In other results, Kaiser's survey found that the top health story followed by Americans in June were articles about the rising costs of health insurance premiums. A total of 83 percent of the public has seen such stories.
Most, if not all of those news stories in recent months have been about the proposed price increases for individual health plans sold on Obamacare marketplaces operated by the federal government and state governments.
Despite that, the survey indicated that half of the respondents said the stories have been about all insurance plans, both individual plans and employer-based plans. Sixteen percent said the reports were about only employer-based plans. Just 10 percent of respondents correctly said the stories related to individual Obamacare plans.