Could the immigration fight cause another government shutdown?

As all signs point to President Obama taking robust executive action on immigration reform before the end of the year, Republicans in Congress are promising an all-out brawl in protest.

And while GOP leaders say they don't want to shut the government down if the president acts unilaterally on immigration, most are not taking anything off the table.

South Dakota Sen. John Thune, a top Senate GOP leader, said Sunday that a government shutdown "doesn't solve the problem," but he acknowledged Republicans are "having those discussions."

Republicans "are looking at different options about how best to respond to the president's unilateral action, which many people believe is unconstitutional, unlawful action on this particular issue," Thune said on "Fox News Sunday."

While the administration has been coy on specifics, the president's actions could shelter millions of undocumented immigrants currently living in the country from the threat of deportation.

Spoiling for a fight, some conservatives have pressed their leaders in Congress to prohibit funding for new green cards and visas in a spending bill that must be passed by mid-December to prevent a government shutdown.

House Speaker John Boehner vowed Thursday to fight the president "tooth and nail" if he takes executive action on immigration, saying "all options are on the table" as the GOP plans a response.

Mr. Obama said Sunday at a news conference in Australia that the threat of a shutdown would not affect the timing of an immigration announcement.

"The main concern I have is to make sure we get it right," he said.

Some Republicans tried to preemptively shift the blame for a shutdown onto Mr. Obama, blaming the president for goading Republicans into a fight.

"The president shouldn't shut down the government so that he can break the law," Gov. Bobby Jindal, R-Louisiana, told NBC News on Sunday.

Jindal, who's mulling a presidential bid in 2016, said Republicans "shouldn't shut down the government," but he said they should "do everything they can to force the president to follow the law."

2012 Republican presidential nominee Mitt Romney told CBS News' "Face the Nation" that Obama would be "poking Republicans with a stick in the eye" if he moves forward alone on immigration.

And Rep. Tom Cole, R-Oklahoma, accused the president of trying to "bait" Republicans into an "extreme" reaction.

"I think he wants a fight," Cole told ABC News on Sunday.

Congressional Democrats, for the most part, have given the president the green light, saying House Republicans had more than enough time to pass a bill after a bipartisan group of Senators passed their own bill last June.

"I have given up on Mr. Boehner on this issue," Illinois Sen. Dick Durbin, the number two Senate Democrat, told CNN on Sunday. "If he wants to step forward and make some explicit promise that the House of Representatives is going to move on comprehensive immigration reform now, while we're in this lame-duck session, then it's another story. Without that, the president should move."

"Boehner won't bring it up because he wants to protect his hard right wing extremists," added Sen. Sheldon Whitehouse, D-Rhode Island, on "Fox News Sunday." "The real story here is the speaker who won't pass bipartisan immigration reform. And we have a broken system. The president has to do something."

Sen. Claire McCaskill, D-Missouri, told "Face the Nation" she's not "wild" about the prospect of executive action, but she said Boehner could prevent it easily enough.

"All he has to do next week, if he doesn't want the president to act, is take up the Senate bill, amend it, change it, put up your own bill," she said. "Let's get back to doing our work instead of just blaming the president for everything."