Washington (CNN) Senate Majority Leader Mitch McConnell is known as the deal-maker on Capitol Hill, but the current logjam over funding the Department of Homeland Security has everyone wondering if he can prevent a shutdown.

Congress returns on Monday with just four days before the deadline to pass legislation to keep DHS open. A stalemate between Republicans and Democrats on Capitol Hill over President Barack Obama's executive actions on immigration has stalled a funding bill for the agency.

Without any serious negotiations or talk of a compromise, it's likely DHS will run out of money at midnight on Friday and the blame game will begin.

Both sides agree Homeland Security shouldn't be interrupted, but the GOP-controlled House passed a measure that funded he agency but also blocked the administration's immigration policies -- something Senate Democrats won't support.

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Congressional Republicans are trying to keep the focus on a key group of six to eight Senate Democrats who criticized the President's move last November to approve policies to allow 5 million undocumented immigrants to avoid deportations. On Friday, the White House spokesman pointed the finger squarely at the GOP.

"The President believes that the Congress, particularly Republicans in Congress, who now have the majority in both the House and the Senate, should fulfill their responsibility to ensure that the Department of Homeland Security doesn't shut down at the end of this month," White House press secretary Josh Earnest said.

Congress has faced showdowns before -- on government funding, on raising the nation's debt limit. The difference now is that nobody is talking; there is no evidence of any back channel negotiations or efforts to float a compromise that might give each side some kind of face saving measure to avoid a shutdown.

McConnell will try a fourth time to start debate on the House bill on Monday, but the vote is expected to fail. The Kentucky GOP leader, who has earned a reputation as a savvy legislator, hasn't signaled whether he has a new strategy up his sleeve.

The spotlight this week will remain on McConnell, who vowed when Republicans took control in 2015 there would be no more government shutdowns. But he's kept mum on how he'll keep that promise.

Last week, a Texas federal judge temporarily suspended the President's immigration actions, a development some thought could provide an escape hatch for Republicans. With the court stopping the program from moving forward some thought the GOP might be more willing to approve the DHS funding as the legal battle played out in the courts.

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But the court's decision appears to have hardened the posture of conservative Republicans, who argue the court backed up their view that the President's actions were unconstitutional.

"Why in the world would we move from that position?" Ohio Rep. Jim Jordan asked CNN in a phone interview.

Jordan heads a new group of House conservatives called the "House Freedom Caucus." The roughly 35 members of that group held a conference call during last week's recess and Jordan reported the group was united in "standing firm."

Members on the right say they remain committed to wage the fight against the administration's immigration policies on this spending bill. Republican leaders convinced members late last year that a possible pre-Christmas fight that threatened a shutdown wasn't the way they wanted to start out a new Congress. Instead they approved a spending bill for all government agencies for a year, but only approved short term money for DHS.

These conservatives are holding House Speaker John Boehner to his pledge to fight "tooth and nail." Any signal of backing down by the speaker would certainly trigger a backlash from many in his conference, as well as grassroots activists.

In an interview on Fox last Sunday, Boehner said he was willing to allow DHS funding to lapse, and repeatedly argued it was up to Senate Democrats to act.

Some House Republicans have told CNN they aren't convinced the legal route will ultimately block the President's immigration actions so they are resolved to keep moving forward on the legislative track.

After the political damage Republicans suffered after a 16-day government shutdown in 2013, Democrats feel confident they can force the GOP to ultimately approve a bill without the immigration provisions.

CNN poll released last week found that a majority of Americans -- 53% -- would blame Republicans on Capitol Hill if DHS ran out of money.

Still McConnell and Boehner are united in their strategy to try to place the blame on Senate Democrats. Despite that multiple Democratic aides say their members are expected to stand together and insist that the GOP relent and pass a clean funding bill.

"Each time that they do the same thing over and over again the caucus gets more united," one senior Senate Democratic leadership aide told CNN.

GOP Sens. Mark Kirk of Illinois and Jeff Flake of Arizona have both said it's time to approve the DHS money and deal with the immigration debate separately, and there are a small number of moderate GOP members in the House that say they could vote for a clean bill. But few believe Republican leaders would completely fold and allow a vote on a clean bill.

It's possible that the Senate could opt to punt the debate for a few weeks or months, and take up a short term spending bill that would keep the agency funded. Republican leaders could point to the Texas court's ruling as effectively blocking the same policies they are trying to prevent in the DHS bill.

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Homeland Security Secretary Jeh Johnson criticized the idea of just passing a stopgap bill.

"Being on a continuing resolution is like driving across country on five gallons of gas at a time. And you don't know when the next gas station is," he said Thursday on CNN.

If the agency funding falls short at the end of the week the majority of DHS' employees -- screeners at airports, security agents at the nation's borders, disaster response teams and secret service agents -- would still have to report to work, but they will not get paid on time.

For now, Republicans aren't worried about the political fallout of a shutdown. Even though most GOP members admit the 2013 government closing was bad for the party they say voters didn't penalize them at the ballot box in the midterm election. The party won control of the Senate and expanded its majority in the House.

Conservatives see this battle as more than one about immigration. They want to set a marker down that they will not accept Obama's efforts to circumvent the authority of the legislative branch.

"If the President can in fact get away with this kind of action -- this executive amnesty action -- which so many people have said violates the separation of powers, what's next? Will this summer will he raise the debt ceiling without a vote of Congress?" Jordan asked.