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Concerned their party is painting itself into a corner, some Republicans are trying to find a way out of the congressional showdown over a House-passed bill that would keep the government open past Sept. 30 only if lawmakers agree to defund ObamaCare.

Senate Democratic Leader Harry Reid on Monday is expected to start the process of setting up a test vote on the House bill.

Reid and his fellow Democrats, naturally, want to strip out the provision defunding the health law. Republicans don't want that. So Tea Party-aligned senators like Ted Cruz have rallied around an only-in-Washington kind of approach -- in order to defend the House bill which they supported, they will try to block Reid from calling it up.

But this raises a perplexing question: Even if Republicans can muster the votes to block the bill, what then?

The approach threatens to end in a stalemate, with the Senate holding on to a bill and neither chamber, then, voting on anything that would keep the government open past Sept. 30. Congress hasn't passed a bona fide budget since 2009, forcing the government to operate on a series of short-term spending bills -- this practice sets up periodic deadlines, and opens the door each time to the risk of a shutdown.

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Some Republicans and Democrats are now looking for a resolution, worried about the political blowback from a shutdown.

Sen. Pat Toomey, R-Pa., suggested Monday that his fellow Republicans set the bar a bit lower, aiming instead to repeal "some of the really egregious things" in the law.

Speaking with Fox News, Toomey said the widely unpopular medical device tax -- which medical device companies warn could put them out of business -- is a prime candidate for repeal. Plus, he said lawmakers should continue to push for a delay in the individual mandate, which is expected to require individuals to purchase health insurance starting in 2014.

"I don't think we're gonna be able to completely defund ObamaCare as long as President Obama's in the White House," Toomey said.

Toomey's comments speak to the divide in the Republican Party. Many GOP lawmakers want to repeal ObamaCare entirely, warning that it is driving Americans off their current health care plans and, because of quirks in the law, forcing others into part-time work. But many also say that, with Democrats controlling the Senate and White House, defunding it is simply not possible.

On CBS' "Face the Nation," Sen. Joe Manchin, D-W.Va., made a suggestion similar to Toomey's.

"Let's fix it. Let's repair it," he said of the law.

Sen. Tom Coburn, R-Okla., on the same show, was skeptical about the GOP push to defund the law. "We don't have the ability ... to put a total stop and defund ObamaCare. It would be nice if we did. I'd be in the fight," he said.

He predicted that Reid would ultimately marshal the votes to strip out the ObamaCare language and send the budget bill back to the House. He predicted the government would not shut down over this "exercise."

With time running out to reach an agreement, there appear to be a few possible outcomes.

Cruz and his allies could successfully stall the bill in the Senate, forcing concessions or forcing a shutdown come Oct. 1.

Or, Reid could succeed in calling up the bill, stripping out the ObamaCare language and sending it back to the House. Then House Speaker John Boehner would be forced to decide whether to stick by his party's demands to defund ObamaCare, or call up the clean budget bill, relying on perhaps a majority of Democrats to pass it.

Cruz, in an interview with "Fox News Sunday," also floated an alternative route. He said the House should "hold its ground," and start passing miniature spending bills, "one department at a time."

"Fund the military, send it over, and let's see if Harry Reid is willing to shut down the military because he wants to force ObamaCare on the American people," he said.

While Republicans debate internally over strategy, Democratic leaders have a largely united message.

Reid said last week that any bill that strips funding from ObamaCare is "dead."

President Obama says any push to nix the health care law is "not going to happen."