Conservatives saw it as raising a white flag when Republican congressional leaders pledged not to withhold funding for the Department of Homeland Security in the fight over President Obama’s deportation amnesty, stoking fears that for the next two years House Speaker John A. Boehner and Senate Majority Leader Mitch McConnell will consistently surrender.

For the Republican base, Mr. Obama’s unilateral move to grant legal status and work permits for up to 5 million illegal immigrants was an unlawful power grab that created a constitutional crisis.

If Republican leaders were not willing to use Congress’ power of the purse — the most potent weapon possessed by lawmakers to restrict a president — to stop a brazen unconstitutional act, conservatives reasoned, would the GOP-controlled Congress ever go to the mat to fight Mr. Obama?

The conservative core of the party has long been leery of Mr. Boehner and Mr. McConnell, who are viewed derisively as establishment stalwarts. The refusal of the leaders to threaten a government shutdown or even close a single agency to force Mr. Obama to revoke his immigration edicts seemed only to confirm the right wing’s worst suspicions.

“The anger I see from my audience at the Republican Party cannot become any more palpable,” nationally syndicated talk radio host Steve Deace said. “We have a president who looks for new and unique ways to shred the Constitution on an almost daily basis, and we have a Republican Party leadership that refuses to do anything about it.”

He said Republican incumbents should expect a backlash and primary challenges next year because of their weak attempt to stop the amnesty.

“People are this angry about it. They feel as if they are already not represented and essentially they have been betrayed by most of the people they just worked to elect in November,” said Mr. Deace. “That’s why people are angry at this, because they realize the people that are in charge of our party don’t believe in almost anything in our party platform. They don’t. They are just treacherous.”

Mr. Boehner and Mr. McConnell declined to use the power of the purse to try to stop deportation amnesty when they pushed through a spending bill in December that funded all of the government except Homeland Security for the remainder of the fiscal year that ends Sept. 30.

They promised a showdown over immigration before temporary funding for Homeland Security expires Feb. 27. But faced with a spate of terrorism in Europe and cyberattacks by North Korea, the Republican leaders also promised not to shut down Homeland Security and jeopardize the safety of Americans.

Meanwhile, Mr. McConnell and his team expressed doubts about getting the 60 votes in the Senate needed to approve the House-passed spending bill for Homeland Security that included policy riders blocking Mr. Obama’s immigration moves.

Republicans have a 54-member majority in the Senate, forcing them to secure support from at least six Democrats to clear the 60-vote bar for advancing most legislation in the chamber.

The pessimism extended to the House chamber.

“We can only do what we can do,” said Rep. Andy Harris of Maryland, though he vowed that Republicans would keep trying. “The fact is, we don’t have 60 votes in the Senate.”

With Republicans bracing for defeat, Democrats intensified their calls for a “clean” spending bill for Homeland Security.

Rep. Louie Gohmert, a Texas Republican who unsuccessfully challenged Mr. Boehner in the speaker election because of the immigration issue, lamented that his party’s leaders had forsaken the only leverage they had when they ruled out a government shutdown in December.

“It’s a real problem,” Mr. Gohmert said.

Republican leaders and their supporters on Capitol Hill insist that they cannot win a shutdown showdown because liberal-leaning mainstream media would perpetuate the narrative that Republicans are to blame for the difficulties and inconveniences of shuttering government offices.

They said that was what happened when Republicans attempted to defund Obamacare and shut down the government for 16 days in October 2013.

The Republican leadership team also points out that the 2013 shutdown didn’t stop Obamacare and a shutdown of Homeland Security this year won’t stop or even slow down Mr. Obama’s immigration policy because agents of work permits — like more than 90 percent of Homeland Security employees — are designated as essential personnel who work even during a shutdown.

“They’ll keep stamping work permits just as fast during a shutdown,” a Republican leadership aide said.

Those pushing for a Homeland Security shutdown, however, are using the same argument.

They say the agents who protect the U.S. are all essential personal and would work through a shutdown, invalidating claims that cutting off funding makes America less safe.

Tea party champions such as Sen. Mike Lee of Utah insisted that they are not giving up on the House-passed bill until the Senate votes it down, which at the very least will put all senators on record of either supporting or opposing the amnesty.

After that, Republicans will have to come up with a way to confront Mr. Obama.

“I’m not going to speculate about that. We’re going to have a plan,” said Senate Majority Whip John Cornyn, Texas Republican.

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