As evidence mounts that Republicans are taking most of the heat for the government shutdown, some conservative backers of the party are breaking ranks over the wisdom of using the budget fight to defeat Obamacare.

Some conservative advocacy groups, such as Heritage Action for America, continue to support attempts by conservative lawmakers, led by Sen. Ted Cruz, R-Texas, to use the pressure of a government shutdown to defund the president's signature healthcare law, The New York Times reports.

Others, however, such as Americans for Prosperity, and the billionaire Koch Brothers, oppose the defunding strategy, believing it to be futile and a backbreaker for the GOP.

A better approach, they say, involves electing more conservative lawmakers over the coming years on the back of opposition to the law which would ultimately lead to the law's repeal, while in the meantime focusing efforts on spending cuts, the Times reports.

"Whether you like it or not, it's the law. And if you don't like it, elect people who can repeal it," Stanley Hubbard, a Minnesota-based television executive and donor to conservative groups, told the Times. Heritage Action, however, says defunding is the best available option to overturn Obamacare.

"If there is a better strategy than defunding, we are all ears," Dan Holler, spokesman for Heritage Action, told the Times. "If it is more workable, sign us up. But nobody was able to present one that would work before October 1st [when the government shutdown began]."

The divisions within the party in general reflect rifts among House Republicans as the budget impasse is about to enter a third week.

Notably, GOP strategist Karl Rove weighed in against the defunding strategy in his Wall Street Journal column in September, foreshadowing what has become a big problem for Republicans.

Though he called the effort "praiseworthy," he said it was also "ill-conceived" and warned that it could have an adverse effect on the party's chances next year in the mid-term elections.

"A shutdown now would have much worse fallout than the one in 1995," he said.

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