WASHINGTON — A partial agency shutdown looming, Senate Republicans offered on Tuesday to permit a vote on Homeland Security funding legislation stripped of immigration provisions backed by conservatives but strongly opposed by President Obama and fellow Democrats.

“We could have that vote very quickly,” Majority Leader Mitch McConnell, R-Ky., as his party struggled to escape a political predicament of its own making involving an agency with major anti-terrorism responsibilities.

McConnell said he did not know how the Republican-controlled House would respond if a stand-alone spending bill passed and the next step was up to the House. Underscoring the realities of divided government, Senate Democratic leader Harry Reid of Nevada initially said he wouldn’t agree to the proposal unless it had the backing of Speaker John Boehner.

With many House Republicans still returning to Washington after a weeklong vacation, Boehner’s office issued a statement that neither accepted nor rejected the proposal that McConnell outlined after weeks of gridlock.

“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?” said a spokesman, Michael Steel.

Senate Republican officials said McConnell’s offer of a vote on a stand-alone funding bill also envisions a vote on a separate measure to repeal a directive from Obama last fall that shields about 4 million immigrants from deportation even though they live in the United States illegally.

At the same time, the proposal would eliminate an attempt by the House to repeal an earlier presidential order that allows tens of thousands of immigrants to remain in the country if they were brought here illegally as youngsters by their parents.

The maneuvering occurred as the president’s party raised the specter of terrorism and the Republicans countered that it was the Democrats who were preventing an orderly renewal of funding for the Department of Homeland Security.

At a news conference a few hours before McConnell spoke, Sen. Amy Klobuchar, D-Minn., urged Republicans to “fund our security and not to send a message to al-Shabab that we’re going to shut down Homeland Security.”

Klobuchar’s state is home to the Mall of America, an enormous facility that was singled out as a potential terror target in a video released by Al-Shabab, an Islamic militant group linked to the al-Qaida terrorist network.

DHS Secretary Jeh Johnson has urged anyone considering a trip to the shopping mall to be particularly careful.

For their part, Republicans sought to pin the blame for a long-standing standoff on Democrats, pointing out they blocked Senate action four times on the combined funding-immigration bill.

They also accused Reid of shifting his demands after first seeking a vote on a stand-alone spending bill, then refraining from accepting McConnell’s offer.

In response, Democrats seemed to backtrack, suggesting they may well sign on to McConnell’s offer whether or not Boehner does.

But even Republicans said privately that they needed to put an end to a controversy that was likely to turn out badly for them.

The current stand-off dates to last fall, when Boehner told fellow Republicans they should allow the funding of Homeland Security without conditions until after the elections. By then, he said, Republicans would have more leverage to force a rollback in the president’s immigration policy.