Senate Democrats filibustered the Homeland Security spending bill Tuesday even before it reached the floor for debate, leaving the department in danger of a partial shutdown this month as Congress fights over President Obama’s deportation amnesty.

Republican leaders, who have said they will not let the department run out of money, now must scramble for a Plan B before a Feb. 27 deadline.

Democrats have said they won’t accept anything other than full funding without any strings attached.

The department has been operating on stopgap funding for four months, and the $40 billion bill was supposed to give certainty through September. House Republicans, however, complicated matters when they attached two amendments canceling Mr. Obama’s 2012 and 2014 amnesties.

Republicans said they would be willing to have a debate on those amendments, but Democrats would have to allow the bill to come to the floor first.

“The only way to finish a bill is to start a bill, and today they voted to refuse to start that process,” said Sen. John Cornyn, Texas Republican.

Senate Republicans have set up another test vote this week, but Democrats said they were wasting their time because they will allow only a “clean” bill.

“We all know this is going to end with a bill funding Homeland Security that goes to the president. They’ll wind up passing a clean bill,” said Senate Minority Leader Harry Reid, Nevada Democrat. “Why do we wait? Why do we agonize?”

The filibuster was backed by all 46 members of the Democratic Caucus and Sen. Dean Heller, Nevada Republican, who said in a statement highlighted by Democrats that he didn’t support Mr. Obama’s actions but he believed the House Republican bill “complicates the process of finding a solution.”

Mr. Obama also has said he would veto any bill that tries to reverse his amnesties, which could grant tentative legal status to up to 5 million illegal immigrants, along with work permits allowing them to compete legally for jobs.

On Wednesday, the president will host six illegal immigrant Dreamers who gained status under Mr. Obama’s 2012 amnesty as a way of putting human faces on those who could be eligible for deportation if the president’s policies are halted.

Mr. Obama chided Republicans for forcing the debate and called the immigration fight “a separate disagreement” that has no place in the Homeland Security bill, which funds cybersecurity, the U.S. Secret Service and the Coast Guard, in addition to border security, interior immigration enforcement and U.S. Citizenship and Immigration Services, the agency that will approve amnesty applications.

“The notion that we would risk the effectiveness of the department that is charged with preventing terrorism, controlling our borders, making sure that the American people are safe, makes absolutely no sense,” Mr. Obama said.

Most of the department’s employees are deemed essential, which means immigration agents, airport screeners and Secret Service personnel would remain on the job even past the deadline, though they would not receive paychecks until the impasse is resolved.

Homeland Security Secretary Jeh Johnson said the funding fight jeopardizes grants that his department pays to states and localities to strengthen security programs.

Both parties denied the fight was over immigration.

Republicans said it was a protest of Mr. Obama’s executive actions. They noted that the president said repeatedly that only Congress could act to halt deportations, before he reversed his position in November and claimed those powers.

Democrats said they refused to let immigration become the focus of the homeland security debate and pointed to the Islamic State’s growing reach in the Middle East and to lone-wolf plots in the U.S., Canada and France as reasons why money must be approved quickly.

The fight could become moot in short order. A federal judge in Texas is expected to issue a ruling any day on a challenge by more than two dozen states that have argued Mr. Obama’s actions violate the Constitution and federal law.

Mr. Obama waited until after midterm elections to announce an amnesty that expands his 2012 policy for Dreamers and creates a category for illegal immigrant parents whose children are either citizens or have legal permanent residency.

Some conservatives tried to force a fight over the policy in the lame-duck session of Congress last year, but Republican leaders pushed the fight until this year. They kept the homeland security spending bill separate from a massive year-end spending bill just to preserve the chance for another fight.

House Speaker John A. Boehner of Ohio said Tuesday that he and his fellow Republicans have done their part and called out Sen. Ted Cruz of Texas and Sen. Jeff Sessions of Alabama as some of the conservatives who have pressed the battle.

“We won this fight in the House,” he said. “Now, the fight must be won in the United States Senate. It’s time for Sen. Cruz and Sen. Sessions and Senate Republicans and Senate Democrats to stand with the American people and to block the president’s actions.”

Mr. Sessions called Tuesday’s failed vote “only the very beginning.”

“This is Day One. What is needed now is a sustained, organized, unified Republican effort to rally the nation against an imperial edict erasing our immigration laws,” he said.

Democrats appear more unified than Republicans.

A number of Democratic senators were critical of Mr. Obama’s executive actions last year but voted Tuesday for the filibuster that upheld his amnesties.

Sen. Joe Manchin III, a West Virginia Democrat who was critical of Mr. Obama, said his vote was intended to send a signal that the homeland security bill wasn’t the place for a fight.

“I stand ready to work with my Republican colleagues to secure our border and reform our immigration system, but this bill is not the proper place to do that,” he said.

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