Could immigration cause shutdown?

• NEW: Reid has privately conveyed concerns to the White House


• Republicans are debating how to confront Obama over immigration


• Some GOP lawmakers say leaders shouldn't rule out a shutdown


• Boehner told GOP he doesn't want a shutdown

Washington (CNN) -- Congressional Republicans are vowing to fight President Barack Obama's plan to make immigration changes through executive action, but they are struggling with how to do that without triggering another government shutdown.

The House and Senate need to pass a measure during the lame duck session to fund federal agencies, which will run out of money in mid-December. House conservatives are pressing Speaker John Boehner to attach language to the spending bill that would block any money for federal agencies to give out any new visas or green cards.

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But House Appropriations Chairman Hal Rogers, R-Kentucky, who is crafting a yearlong so-called "omnibus" spending measure, warned that effort could trigger a shutdown, and insisted no one wants to go down that road.

"There's no one stronger than me against unilateral action by the President on this subject, however it's been said before - don't take a hostage you can't shoot," Rogers told reporters after a meeting with all House Republicans on Thursday.

The debate over how to confront Obama is posing a sudden challenge for Republican leaders fresh off their victories during last week's midterm elections, which gave them control of both chambers of Congress for the first time in nearly a decade. A day after the election, Mitch McConnell, who will become Senate majority leader in January, said Republicans would take positions Obama wouldn't like but pledged not to shut down the government.

Obama, who is traveling in Asia this week, has said he remains committed to using his executive authority to address immigration by the end of the year but the precise timing isn't clear. White House press secretary Josh Earnest told reporters in Myanmar that Obama is "nearing a final decision."

Boehner told members at Thursday meeting that one of his top priorities is to use every tool available to him to stop any effort by the president to go around Congress on immigration, according to several Republicans who attended. But the Speaker also said he doesn't want a shutdown, and he cautioned members that they need to game out a smart way to respond before they move forward with a vote.

At an afternoon press conference, Boehner pledged to "fight the president tooth and nail if he continues down this path."

He said no decisions were made and GOP leaders were considering various ways to block the action, but did not rule out waging the fight on the spending bill.

"All options are on the table," he said.

READ: Obama determined to bypass Congress

Arizona Republican Rep. Matt Salmon, who drafted a letter signed by more than 50 House GOP members to Rogers, told reporters he's not pushing for a shutdown, but leaders shouldn't rule out any options.

"It's really tragic that before we even fight a fight, or even stake a claim that we're throwing in the towel. I just don't think that's the way to operate. We don't have to talk about ultimatum scenarios at this point in time," Salmon told reporters after the meeting.

Salmon argued that after the midterm elections Republicans have a mandate to push back at the president, and they should use their legislative authority to curb his executive actions.

Senate Majority Leader Harry Reid supports Obama's effort to address immigration but has privately conveyed concerns to the White House that an executive action could threaten passage of the spending bill.

"The President has said he'll do the executive action, the question is when," Reid told CNN in a hallway interview in the Capitol. "He can do it, you know, it's up to him. But I'd like to get the finances of this country out of the way before he does it. But it's up to him."

Several Republicans told CNN the memory of last year's shutdown and the damage it inflicted on the GOP brand is fresh enough that members are willing to give the Speaker some time to decide what approach to take. A vote on a spending bill is not expected on the House floor until early December.

READ: Harry Reid: Dems won't engage in 'obstruction'

What's complicating the next move for Boehner and other congressional Republican leaders is no one knows the details yet on the timing or scope of what the President will do.

Even if House Republicans have the votes to pass something as part of a spending bill, any measure that blocks the president from using his own authority will be blocked in the Senate, which is still controlled by Democrats, or vetoed by the president. That leaves the clock ticking before the government runs out of money on December 11th.

One option GOP leaders are considering is passing a short term spending bill -one lasting a few weeks even - that would avoid a shutdown and give them more time to decide how to respond to the President after he rolls out his plan, which is expected by the end of the year.

Another is to expand the lawsuit against President Obama that House Republicans already approved to challenge his authority to implement Obamacare. Although the House voted to sue the president in July, the formal legal paperwork hasn't been filed with the courts yet.

Whether leaders decide to challenge the president as part of a spending bill or later, it's clear that conservatives are bracing for a major confrontation with the president over his plans to allow potentially millions of undocumented workers to remain in the U.S.

"I trust him completely to do the worst left wing, twisted policy that he can come up with" Arizona Republican Rep. Trent Franks told CNN.

READ: What to expect from the lame duck Congress

CNN Chief Congressional Correspondent Dana Bash and Ted Barrett contributed to this story.