Congressional Republicans' willingness to risk a government shutdown to defund Obamacare could squander their single biggest political asset heading into the 2014 elections: the party's unified opposition to this increasingly unpopular law.

House Republican leaders calmed a brewing intraparty divide this week when they announced support for a budget bill that would keep the government running beyond the Sept. 30 deadline, but eliminate funding for implementation of the Affordable Care Act. The legislation is set for a vote Friday and the overwhelming GOP support it is expected to have should quiet a sharp disagreement over the tactics of defeating Obamacare that has engulfed Republicans for weeks.

But the reprieve is likely temporary. The issue is sure to resurface early next week, when the Democratic Senate takes up the budget bill.

The Senate is expected to strip the Republican bill's Obamacare provision, replace the money for its implementation and return the legislation to the House, putting back in Republican hands the responsibility for passing a budget bill or allowing the government to shut down on Oct. 1.

Neither House Republicans committed to defunding nor pragmatists worried that a shutdown will backfire politically have figured out what to do when the Senate and President Obama inevitably reject the defunding provision and the government shuts down. One option is an alternative House bill that doesn't defund Obamacare but would delay its implementation for a year. Republicans bent on defunding, however, have so far showed little enthusiasm for a delay.

“I think we have a united front, not just among conservatives, but among the majority of our conference, to really fight for this thing,” said Rep. Mark Meadows, R-N.C., a House leader of the defund movement. “Going back on my word, to allow Obamacare to be implemented, is not something that I can do, nor many in our conference can do.”

Polls continue to show low public support for the new health care law, whose implementation will accelerate on Oct. 1. Among Americans' worries is that Obamacare would raise insurance rates and reduce access to quality care.

However, negative opinions of Obamacare have not translated into support for the Republican strategy to shut down the government unless Democrats agree to defund Obama's signature legislative achievement. Every publicly available poll done since the defunding campaign began shows that voters would blame Republicans for a government shutdown far more than Obama or congressional Democrats.

Republicans opposed to the defunding strategy worry voters will resent the tactic and shift their ire from Obamacare to the loss of public services caused by a shutdown. That kind of backlash, opponents fear, would turn a winning, unifying issue into a loser that could cost the GOP House seats and a shot at control of the Senate in the 2014 mid-term elections.

“We’d all like to get rid of Obamacare, and the question is, how?” Sen. Lamar Alexander, R-Tenn., said. “I’m not in the ‘shut down the government’ crowd; I’m in the ‘take over’ the government crowd and let’s replace Obamacare.”

Sen. Ted Cruz, R-Texas, the national face of the defunding campaign along with Sen. Mike Lee, R-Utah, has conceded what most in Washington already know: That Republicans do not have the votes in the Senate to approve the defunding of Obamacare.

The lack of Senate support means it will be up to House Republicans to determine how far they'll press this fight and that's one of the reasons House GOP leaders have offered the alternative proposal to delay the health care changes for a year. Obama has supported delays in other parts of the law, including the employer mandate.

Still, even Republican defunders who tacitly acknowledge that the American people could turn against them in the event of a government shutdown seem unwilling to back down, though that could change over the next 10 days when the prospect of a government shutdown moves from hypothetical to likely. For now, though, they're holding the line.

“I’m kind of in a wait-and-see mode,” Rep. Matt Salmon, R-Ariz., said. “I think the best thing that would happen is we don’t have a government shutdown.”