They just can't agree on how to do it.
Should they shut down the government? File a lawsuit? Or is there a more measured approach that could showcase Republican leadership, such as passing a legislative alternative?
"We're working with our members and looking at the options that are available to us," House Speaker John A. Boehner (R-Ohio) said Friday. "But I will say to you that the House will, in fact, act."
Here's a look at their leading options.
1. Do nothing — for now.
The heaviest lift for Republican leaders would be to persuade their members to take a deep breath and hold their fire. They could launch committee hearings and investigations into Obama's actions — one is set for early next month — but otherwise shift to different priorities. That would mean working with Democratic leaders to pass a so-called omnibus appropriations bill by the Dec. 11 deadline to fund the government until next October, lifting the risk of another shutdown.
Before the president announced his executive action, leaders on both sides supported an omnibus bill, which would let the Republican majorities start fresh in January on their agenda and begin sending legislation to the president.
Incoming Senate Majority Leader Mitch McConnell (R-Ky.) says that by breaking congressional deadlock and pursuing a proactive, not reactive, agenda, Republicans will secure a better footing to rein in the administration's policies.
Conservative lawmakers and allied outside groups call such an approach dead on arrival.
"Inaction is not an acceptable response," said Michael Needham, chief executive of the conservative group Heritage Action, in a statement urging lawmakers to deny funding for the president's program. "Anything less will amount to a blank check for Obama's unlawful amnesty program."
2. Use the budget process.
Even before Obama unveiled his plan, some Republicans wanted to reject the nearly 10-month omnibus bill and agree to only short-term extensions of government funding to keep the lights on until early next year. That would put off fears of a holiday shutdown, but let Republicans retain leverage to try to force Obama to back off his new proposals.
It would also give the party more to gauge public reaction. If Obama's plan proves to be a political flop, there might be greater public support for efforts to cut funding for government immigration programs.
The downside for Republican leadership is that a short-term spending bill forces the party into another round of fiscal fights when they'd rather be advancing legislation on taxes and trade.
A few hard-core conservatives advocate inserting language to halt Obama's immigration plan into the upcoming spending bill, even if it triggers another shutdown. But most Republicans seem to oppose such a drastic step.
Also, Congress may find it difficult to use the power of the purse to halt Obama's deportation deferrals. White House officials say their plan will be funded by fees and fines paid by immigrants and would not be affected by a shutdown.
"How do you stop an inaction? That's the tough question that I don't have the answer to today," Rep. Pat Tiberi (R-Ohio) said, noting Obama's policy in part relies on not deporting certain categories of immigrants. "Shutting the government down — that doesn't stop this inaction. Don't fund immigration services — that doesn't stop this inaction."
Republican leaders are also working quietly on a possible hybrid approach: an omnibus bill that funds everything except immigration-related operations through the fiscal year but that allows money for immigration programs only through the holidays.
The approach would avert a shutdown but set up a fresh debate on Obama's immigration policies early next year, when Republicans control Congress.
3. Censure or impeach Obama.
The I-word, as in impeachment, has been mentioned as a possible option — but more often by Obama supporters who cite the threat as an example of Republican overreaction. That's not to say Republicans haven't floated the idea, or at least refused to rule it out.
Rep. Steve King (R-Iowa) says it would be better to censure the president. Such a rebuke would be rare, and largely symbolic. The only president to have been censured, Andrew Jackson, later had the action expunged.
King also says the House could pass a resolution of disapproval of the new policy, which would be a similarly symbolic gesture and could be done relatively soon.
4. Go to court.
House Republicans filed a lawsuit Friday against the president for failing to enforce part of the Affordable Care Act, which they cite as an example of executive overreach. Boehner has not ruled out expanding that suit to include the executive action on immigration.
Even senators who supported a 2013 bipartisan Senate immigration bill, including Sen. Susan Collins (R-Maine), prefer this option. "Probably the best approach is to challenge the president in court," she says.
But legal experts say such lawsuits are typically rejected by the courts, which tend to stay out of power struggles between Congress and the White House.
5. Pass an immigration bill.
Obama and other Democrats have said Republicans who oppose his decision to take executive action could address those concerns by passing legislation of their own. But that's unlikely given the divisions inside the GOP.
After the Senate bill on immigration passed last year, the House refused to consider it, preferring to take what Boehner has called an incremental approach. Discussions among House Republicans that began in earnest after Obama's reelection in 2012 largely fell apart when House Majority Leader Eric Cantor (R-Va.) lost his reelection primary race in June.
But leaders say they may yet take up immigration in the new Congress.
"Let's start moving immigration legislation that we like," said Rep. Tom Cole (R-Okla.), listing bills on border security, HB-1 visa reform and seasonal labor as possibilities. "We should start picking the things that are important and see if the president wants to veto those things. I think it'll make it a lot clearer who's trying to work to a solution and who's not."