Senate Majority Leader Mitch McConnell (R-Ky.) speaks to reporters following the weekly Republican policy luncheon Tuesday. (Mandel Ngan/AFP/Getty Images)

The Senate moved closer Tuesday to a deal to avert a partial shutdown of the Department of Homeland Security, but the proposal faced an uncertain future in the House, where Republican leaders conspicuously refused to embrace it.

Senate Majority Leader Mitch McConnell (R-Ky.) told reporters he was prepared to move swiftly to extend funding for DHS through the fiscal year in a bill that is not contingent on Republican demands to repeal President Obama’s executive actions on immigration.

Under McConnell’s proposal, the Senate would vote first on the funding measure and then hold a separate vote on a bill to undo Obama’s new immigration initiatives. McConnell hopes to assuage conservatives who are determined to confront the president on what they see as abuse of his executive authority.

“I don’t know what’s not to like about this,” McConnell said. “This is an approach that respects both points of view.” If successful, the proposal will break a two-month deadlock over funding for the agency that is responsible for border security, airport security checks and a range of other functions.

But House Speaker John A. Boehner (R-Ohio) did not immediately warm to the proposal, and it was not clear whether he could marshal enough backing in his chamber to complete the deal to keep DHS open beyond Friday, when its spending authority expires.

Asked about the possibility of a shutdown of the Department of Homeland Security during a news conference on Wednesday, Speaker of the House John Boehner repeatedly urged the Senate to act on the issue. (AP)

House Republicans will huddle behind closed doors Wednesday morning. The unsettled DHS debate is expected to be the central focus of their discussion.

The stalemate has tested Republicans’ ability to govern now that they are in full control of Congress. Senate Democrats have blocked four attempts by McConnell to move forward on a House bill that would fund the department but that ties that funding to a repeal of the president’s immigration actions, which would allow millions of undocumented immigrants a temporary reprieve from deportation.

Again on the brink

McConnell’s move represented a concession in a fight that has threatened to shut down the agency less than a year and a half after a budget standoff shuttered a broad swath of the federal government for more than two weeks in October 2013.

Senate Minority Leader Harry M. Reid (D-Nev.) said Democrats want assurances from Boehner that a “clean” funding bill will pass the House before they will support McConnell’s plan and allow the votes to move forward.

“Now, all eyes are on Speaker Boehner,” said Sen. Charles E. Schumer (D-N.Y.).

Asked about the emerging Senate plan, Boehner spokesman Michael Steel said by e-mail: “The speaker has been clear: The House has acted, and now Senate Democrats need to stop hiding. Will they continue to block funding for the Department of Homeland Security or not?”

Exiting a House leadership meeting later on, House Rules Committee Chairman Pete Sessions (R-Tex.) said he did not support approving McConnell’s plan. Instead, Sessions said, Congress should pass a temporary extension of funding for up to six weeks and convene a House-Senate conference to try to hammer out the differences between the two chambers.

DHS is scheduled to begin furloughing nonessential workers if Congress does not extend its $40 billion budget by Friday.

At the White House, Obama prepared to inject himself more forcefully into the debate with a town-hall-style forum planned for Wednesday in Miami, which has a heavily Latino population. The president is expected to address the funding standoff and his immigration plan. The administration has appealed a federal judge’s decision last week temporarily blocking the deferred deportations program, under which up to 5 million of the nation’s 11 million undocumented immigrants would be eligible for renewable three-year deportation waivers. Many could receive work permits.

As the president tries to rally the public, however, Obama and his aides have limited their dealings with Capitol Hill. White House and Democratic aides said there is little the president can do, because he will not undo his immigration actions. It is up to Republicans to decide whether they are willing to shut down DHS over the issue, the aides said.

“This is not a battle with the White House. This is a battle taking place on Capitol Hill,” a White House official said, speaking on the condition of anonymity to discuss internal thinking. “Legislators on the Hill need to work on this. . . . What you’re going to see from the administration on DHS up until Friday is continuing to call attention publicly to the potential impact of shutting down DHS.”

To that end, Homeland Security Secretary Jeh Johnson and two predecessors from the George W. Bush administration — Michael Chertoff and Tom Ridge — will attend a news conference scheduled for Wednesday to warn of problems that could occur if the agency is forced to begin furloughs.

Essential personnel, as many as 200,000 employees, would continue to report to work without pay during a partial shutdown. Although Johnson and the White House have said that national security would not be jeopardized, they have warned that long-term planning could be affected. In an interview, Chertoff warned of morale problems in the agency and said he thinks a shutdown would “no doubt adversely affect the nation’s security.”

“A couple days of delay [of pay] will not make a difference, but if it starts to mount up and there’s real uncertainty, people will start to feel demoralized, and that could have an impact over a long period,” Chertoff said. “You wouldn’t do this to [military] troops in the field, send them into combat but not get paid. They need to take this seriously.”

Doubts on the right

While McConnell’s plan to split off the immigration issue into a stand-alone measure opened the door to winning over Democrats, conservatives were skeptical.

“Senators arguing fund DHS but vote a separate bill to defund executive amnesty. Have you heard of Obama veto? Think we were born yesterday?” Rep. Steve King (R-Iowa), a staunch opponent of Obama’s immigration actions, wrote on Twitter.

Meanwhile, the administration has struggled to explain how a partial shutdown would affect national security. White House press secretary Josh Earnest, asked repeatedly about it Monday, declined to say that the nation would be more at risk. Rather, he said, the situation would not improve the nation’s safety.

“It’s hard to imagine a good time for Congress to be mucking around with the funding of the Department of Homeland Security,” Earnest said. “But now seems like a particularly bad one.”

Politically, the standoff could help the White House and Democrats. Republicans drew the brunt of public anger over the 2013 shutdown, and administration aides believe the same will happen if DHS is affected at week’s end.

It is not lost on White House allies that the president is traveling to Florida, a key swing state in presidential races, for the Wednesday event, co-hosted by MSNBC and Telemundo.

“In our travels to the immigrant community, the faith community, the law enforcement community and the business community, they want someone in D.C. to do something,” said Ali Noorani, executive director of the National Immigration Forum. “The president has done something. The political risk to the administration is only there if they stop doing something. Right now, it’s on Republicans to kind of meet the bar.”

Senior administration officials are scheduled to appear at another immigration forum, hosted by Univision, on Sunday in Los Angeles, the city with the largest number of immigrants potentially eligible for deportation relief under Obama’s executive actions.

“For the first time in a long time, the immigrant community and advocates are very aligned with the White House,” said Marielena Hincapié, executive director of the National Immigration Law Center. “I think the fact that he’s coming to Miami to meet with immigrant community members and take questions directly from folks speaks volumes.”

Paul Kane contributed to this report.