While all Republicans supported defunding ObamaCare, those aligned with the Tea Party wanted to take the strategy further.
Tea Party voters are the least likely to see a government shutdown as a big problem, according to the Pew survey. Asked about the severity of a shutdown, 61 percent of all respondents and 51 percent of all Republicans said it would have a "major effect" on the economy.
But just 39 percent of Tea Party Republicans predicted a major economic impact, compared with 51 percent who expected only a "minor effect" on the economy.
Most Tea Party voters said the economic effect of a shutdown, whatever its size, would be negative.
Overall, the Pew survey found, Republicans and the White House would share the blame for a government shutdown. Thirty-nine percent said they would blame Republicans, compared with 36 percent who said they would blame the Obama administration.
Republicans who were in office during the 1995 government shutdown have warned that Congress will end up with the blame if the current shutdown threat comes to fruition.
Newer conservative members, on the other hand, say that Obama is likely to "blink first" if the government does shut down.
The strategy would not cut off ObamaCare funding on its own. Rather, conservatives believe the political pressures to reopen the government will lead Obama to sign a bill preventing his administration from implementing the law.