Despite these views, the public generally has a more unfavorable view of the replacement legislation than a favorable one.
His and Johnson's sentiments echo those expressed by a number of Senate Republicans who are wary of the AHCA's massive cuts to Obamacare protections and programs, particularly its expansion of Medicaid.
The House-passed AHCA would eventually lead to 23 million fewer people covered, according to a recent Congressional Budget Office estimate. The poll was before the report's release. It also found that people with pre-existing conditions would be priced out of the market. Others, however, want to hew to a tough line on state flexibility, as well as Medicaid spending reductions and other issues.
In the KFF poll, even Republicans showed scant support for changes to some of the PPACA's most-popular provisions.
McConnell last week said he did not know how the Senate would get to 51 votes on healthcare, and suggested moving tax reform legislation could be simpler.
"In fact, most of the instability driving up premiums in the marketplace can be directly traced to Republicans' efforts to undermine the healthcare law for their own political purposes", he wrote.
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The dramatic quest to repeal Obamacare this spring left House Republicans politically bruised, exhausted and - perhaps most of all - relieved to finally be done.
Wednesday's survey from the nonpartisan Kaiser Family Foundation also finds a growing share of the public concerned that the Republican bill will have negative consequences for them personally by increasing their costs, making it harder to get and keep coverage, or reducing quality.
About 75 percent of those surveyed say the bill passed by the House doesn't fulfill Trump's promises on health care. The shocking ordeal was a sharp reminder of how much difficulty some Republicans are having in defending the House's controversial health care bill - and how important it will be for lawmakers and candidates to be prepared with appropriate responses.
It was the latest in an ongoing series of Kaiser polls on health care. On Thursday, Burr expressed his skepticism during a local TV interview in his home state of North Carolina while Johnson said he wasn't "sure [he] would bet on" a package to repeal and replace the health law. More Americans, interestingly, view Obamacare favorably than the AHCA.
Meanwhile, insurers continue to announce significant rate increases if the cost-sharing reduction (CSR) payments are not made.
The poll was conducted among a random national sample of 1,205 adults; overall results carry a three-point margin of sampling error.