A few prominent conservative groups have begun to incrementally soften their positions against raising the federal debt limit, and appear to be committing more intensely to a fight anchored in a government funding debate.

Heritage Action CEO Michael Needham said at a Christian Science Monitor breakfast Wednesday that while the debt limit should ultimately be raised, Republicans should continue to reject any government funding measure that does not weaken the Affordable Care Act.

"My tactic is to focus on the CR," Needham said, referring to a measure that would reauthorize federal funding and reopen the government, which Republicans have opposed pending concessions from Democrats.

Matt Kibbe, president and CEO of Tea Party ally FreedomWorks, told the Huffington Post on Wednesday that his group is also focused on the government funding debate.

"This is an emerging set of opinions that looks to be where everybody's gravitating to," he said.

The government shutdown, while serious, is widely considered less economically damaging than a failure to raise the debt ceiling, which would force the U.S. to default on its debt payments.