But 44 percent of voters they they would be less likely to vote for a member of Congress who backed the bill, according to that poll from Quinnipiac University.
The same Quinnipiac Poll found that 57 percent of voters disapprove of the bill, known as the American Health Care Act. Just 20 percent of voters approve of the bill.
"Advisory to Republicans who support the replacement for Obamacare: Backing this bill could be very hazardous to your political health," said Tim Malloy, assistant director of the Quinnipiac University Poll. The poll questioned 1,404 voters, and had a margin of error of 3 percent.
The Q-Poll's findings were mirrored by those of another survey, the Harvard-Harris poll, which found that 55 percent of voters see the AHCA as a step backward.
But support for the bill was higher in the Harvard-Harris poll, which found that 45 percent of voters see it as a step forward.
That said, 57 percent of respondents told the poll they want significant changes made to the bill by the Senate. The working group of senators reviewing the bill has indicated that their version will look greatly different from the House version.
Mark Penn, co-director of the poll, told TheHill.com, "The voters want to neither go back to Obamacare nor to the House bill."
"The Senate is going to have to thread the needle here and craft a new compromise," said Penn, whose poll questions 2,006 voters. The poll's methodology does not have a traditional margin of error, according to TheHill.com.
A third poll came from Hart Research Associates, which was commissioned to survey 1,005 voters online by Protect Our Care, an Obamacare-defense group.
That poll found that just 40 percent of voters have a favorable impression of the bill, compared to 54 percent with an unfavorable impression.
However, support for the bill worsened after respondents were told about provisions in the measure that would lead to some older insurance customers and people with pre-existing conditions facing higher health plan premiums, as well as about a big tax cut in the bill for wealthy Americans.
After hearing that, 65 percent of respondents said they saw the bill unfavorably, compared to 35 percent who viewed it favorably, according to the poll, which had a margin of error of 3 percent.
A total of 62 percent of voters said they wanted to see the Senate "work on a new bill that keeps what works and fixes what doesn't in the [Affordable Care Act] rather than repealing the law altogether," according to a summary of the results from Hart Research.
Watch: CBO scores new health care bill