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But even as GOP leaders try to shift the blame to Democrats, some Republicans are starting to break ranks with their leaders' strategy.

House Speaker John Boehner, pressed Thursday if he'd guarantee DHS wouldn't be shut down, sidestepped the question, and instead repeated his mantra: The House had done its job.

READ: Boehner tells Senate Democrats to 'get off their ass'

"We've passed a bill to fund the department and stop the President's unilateral action with regard to immigration," Boehner said in his weekly press conference.

But action is stalled on that bill.

It's "stuck," according to Senate Majority Leader Mitch McConnell, who has failed on three attempts to attract enough Senate Democrats to agree to bring the measure up for debate in that chamber. Democrats are unhappy House Republicans attached several controversial amendments to the bill that would block President Barack Obama's immigration policies. Democrats are insisting they will support only a "clean" funding bill, without the immigration provisions.

McConnell attempted to flip the ball back in Boehner's court earlier this week, but Boehner made it clear he wants to keep the pressure on the Senate.

While Boehner and McConnell appeared to point fingers at one another earlier this week, they are now coalescing around a common strategy: pin the blame on a group of roughly seven Senate Democrats who criticized the President's decision last fall to unilaterally stop the deportations of roughly 5 million undocumented workers who entered the U.S. illegally.

"If funding for Homeland Security lapses, Washington Democrats are going to bear the responsibility," Boehner said.

West Virginia Democratic Sen Joe Manchin, who admitted he disagreed with the President's immigration actions, told reporters he strongly opposes adding the GOP immigration amendments to the DHS bill and he plans to hold firmly against it.

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Republicans struggle to avoid Homeland Security shutdown