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An aggressive push by Tea Party lawmakers to defund ObamaCare is increasingly pitting Republicans against Republicans, as some party leaders and conservative pundits claim the goal is not achievable this year -- and could irresponsibly risk a government shutdown.

Sen. Mike Lee, R-Utah, and his allies in the Senate are trying to round up support for a pledge to oppose any budget bill that funds the health care law. They're emboldened by the administration's recent decision to delay a key part of the law, arguing the move shows it's not ready for prime time.

"Businesses don't like it. Individuals hate it. Union leaders say it will be bad for workers," Lee said on "Fox News Sunday." "The law is certainly not ready to implement, and we shouldn't fund it."

But the demand, if backed by enough lawmakers, could create an impasse in Washington -- risking the possibility that Congress will miss its Sept. 30 deadline for passing a budget and shutting down the government.

A number of GOP lawmakers have come out against the defund-ObamaCare campaign, claiming that while they'd like nothing more than to strip the law of its funding, the party does not have the numbers to deliver.

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"I'd be leading the charge if I thought this would work. But it will not work," Sen. Tom Coburn, R-Okla., told the Washington Examiner.

Coburn did not hold back in criticizing the campaign, accusing fellow Republicans of being "dishonest" with their base.

"It's not an achievable strategy. It's creating the false impression that you can do something when you can't," he said.

Earlier, Sen. Richard Burr, R-N.C., said in an interview with Public Radio International that the shutdown threat is the "dumbest idea I've ever heard."

He and others argue that Congress will not be able to defund the health care law as long as Obama is in the White House.

Yet if enough Republicans sign on to the effort, they could prevent the Senate from advancing a budget bill -- in turn threatening the shutdown.

Washington has been down this road before, most recently averting a shutdown this past March. And both sides of the aisle are flirting with a shutdown threat to get their way this year. While some Republicans claim to be holding out for the health care law to be stripped of funding, some liberal Democrats are beginning to insist that sequester spending cuts be changed or undone as a condition for support on a budget bill.

The administration offered words of support for the latter efforts on Sunday, with Treasury Secretary Jacob Lew saying: "Congress should find a way out of sequester."

On the Republican side, a dozen senators -- including Sens. Marco Rubio, R-Fla.; Ted Cruz, R-Texas; and Rand Paul, R-Ky. -- have so far backed the push to defund ObamaCare.

They began rounding up support after the administration opted to delay by a year the requirement on some employers to provide health care coverage to workers.

"If Democrats will not agree with Republicans that ObamaCare must be repealed, perhaps they can at least agree with the president that the law cannot be implemented as written," they wrote in a letter to Senate Democratic Leader Harry Reid. "If the administration will not enforce the law as written, then the American people should not be forced to fund it."