Sen. John Thune said he's worried that congressional Republicans could squander GOP unity on Obamacare, and the public's dissatisfaction with the law, because of a fresh disagreement over political strategy.

Thune, of South Dakota, is the No. 3 Senate Republican and in charge of the GOP caucus' messaging. He's also the only GOP leader to sign a controversial letter that calls for shutting down the government if President Obama refuses to defund the Affordable Care Act.

But while Republicans are united in their opposition to the president's health care reforms, they've split over how to derail the new law, whose implementation accelerates next year. Some conservative Republicans favor the defund-or-shutdown strategy while GOP leaders and other Republicans warn that it could backfire on the party.

"I think it's important that -- even though there may be differences in people's approach on this and the tactics that are used to get to the goal line -- that we stay united on the goal itself," Thune told the Washington Examiner Tuesday. "And, that from a standpoint of the way we talk about this issue -- that we continue to reiterate that Republicans of all stripes and persuasions are opposed to Obamacare, oppose its implementation [and] want to do everything we can to see that it's repealed or permanently delayed."

The disagreement has played out in public over the past week, with Republicans for and against the defund-or-shutdown strategy taking to media outlets to trade barbs and accuse the other of supporting an approach that is misguided, dishonest, or both. Thune wants it would stop.

"It's important that Republicans not get into the differences of opinion about what's the best way to proceed and have that discussion in the public way," the senator said. "It's important that we, as much as we can, keep focused on the ultimate goal and try to stay together on that, knowing that we, at least for the time being, have some different ideas about what the best way is to proceed."

The Tea Party-affiliated senators behind the letter Thune signed, Republicans Ted Cruz, of Texas; Mike Lee of Utah; and Marco Rubio, of Florida, have suggested that members who do not sign the document and back their effort are "surrendering" without a fight, and aren't truly committed to blocking Obamacare. Republicans who oppose this strategy -- and that includes most -- have mocked the strategy, contending that it is political malpractice and doomed to failure.

These Republicans note the long odds of pressuring Senate Democrats into joining with the House and Senate GOP to block continued implementation of Obamacare -- never mind the unlikelihood of convincing the president to defund his signature legislative achievement. The intraparty squabbling has Thune concerned, and it is why he is counseling all of his GOP colleagues to cool their rhetoric and focus on where they agree.

Despite signing the letter backing the strategy, Thune indicated that he is open to supporting the continuing resolution needed to keep the government open in exchange for meaningful concessions that fall short of defunding Obamacare, but that chip away at the program and delay its implementation. The South Dakotan expects most Senate Republicans to unify around this strategy as Sept. 30 approaches, although he conceded that some Republicans are committed to their current positions.

"I've tried to convey to my colleagues the importance of us being united on message, because this is an issue, for me, where we really have the high ground. Democrats are running for tall grass," Thune said. "Republicans are very well positioned on this, and we should make sure that we don't squander that, that we approach this in the right way and try and minimize the difference about tactics but stay focused on the overall objective."