Senate Democrats blocked legislation Monday that would have rolled back President Obama's executive actions on immigration in exchange for funding the Department of Homeland Security through September.

But soon after the early evening vote -- the fourth Senate attempt to block Obama's controversial decision to grant work permits to millions of illegal immigrants -- Senate Majority Leader Mitch McConnell, R-Ky., suggested separate legislation to combat Obama's executive actions.

"The new bill I described offers another option we can turn to. It's another way to get the Senate unstuck from a Democrat filibuster and move the debate forward," McConnell said on the Senate floor after a vote to advance the House-passed bill failed 47-46, short of the 60 votes needed. Three previous attempts earlier in the month had yielded similar results.

It was not clear whether McConnell's gambit would succeed ahead of Friday's midnight deadline to fund the department or see it shut down. It was far from certain whether it would win any Democratic support, and House conservatives remain firmly opposed to any funding bill for the Homeland Security Department that does not also overturn Obama's executive actions on immigration.

If no funding deal is reached by the deadline, the DHS could partially shut down, resulting in the furloughs of roughly 30,000 DHS employees. About 200,000 others would continue to work, but they would receive no pay until Congress authorizes funding.

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It's a reality that was on display during the 16-day government-wide shutdown in the fall of 2013, when national parks and monuments closed but essential government functions kept running, albeit sometimes on reduced staff.

Earlier in the day, Obama again warned that failing to act before Friday increases the risk of a domestic terror act.

At a White House gathering of governors, Obama accused Congress of creating “self-inflicted” wounds and said failing to pass the funding bill within the next several days “will have a direct impact on America’s national security.” Homeland Security Secretary Jeh Johnson appeared on all five Sunday talk shows to make a similar case.

The Tea Party Patriots group is suggesting that Senate Republicans are backing down because they fear Americans will blame them for a partial DHS shutdown.

“Senate Republicans are about to cave in to President Obama,” the group said Monday. “It’s time … to ratchet up the pressure on wobbly Senators.”

A federal district court judge in Texas last week temporarily blocked the administration's plans to carry out an executive action that protects millions of illegal immigrants from deportation.

The Justice Department on Monday asked a federal judge to lift the judge’s temporary block and make a decision by Wednesday. If the judge fails to rule in the administration’s favor, the department is expected to turn to the 5th Circuit Court of Appeals.

Wisconsin Sen. Ron Johnson, who leads the Senate committee that oversees the DHS, was not calling Monday for the immigration provision to be removed from the funding bill, but suggested that the courts, not Congress, will resolve the issue.

“Now that the judiciary branch is involved, the courts ultimately will decide the constitutionality of the administration’s ‘deferred action’ memorandum,” he said.

Fox News' Kara Rowland and The Associated Press contributed to this report.