"There is nobody who denies this will require courage," said Heritage Action CEO Michael Needham on a call with reporters. "Maybe [Congress's] approval rating is at 12 percent because they haven't tried to inspire people."
The survey represents the latest volley in a struggle between conservative and establishment Republicans on how to handle the fiscal debates that await Congress this fall.
Heritage Action and a long list of conservative groups are supporting a campaign by Sen. Mike Lee (R-Utah) to pass government funding bills in September that do not include money for ObamaCare.
Proponents of the strategy suggest that it will force Obama to choose whether to fund the healthcare law or shut down the government when the current operations bills expire on Sept. 30.
"The only person out there threatening a shutdown is President Obama," Needham said Wednesday.
On the other side are players like House Majority Leader Eric Cantor (R-Va.) and strategist Karl Rove, who argue that passing a funding bill without money for ObamaCare will be next to impossible in the Senate, which is controlled by Democrats.
Unless Lee can muster 14 Democratic votes for bill that defunds ObamaCare, his strategy goes nowhere, Rove said Tuesday.
"This is the one strategy, the one tactic that might be able to guarantee that the Democrats pick up seats in the Congress in 2014," Rove said on Sean Hannity's radio show.
Needham dismissed this claim on Wednesday and predicted that lawmakers will be "rushing" to support the defunding strategy when they get back from August recess.
Heritage Action is holding events in nine cities over the next several weeks in order to put pressure on members of Congress.
The Basswood Research survey was conducted from 1,000 likely voters in 10 Republican-leaning House districts. It has a margin of error of about 3 percentage points.