WASHINGTON – House Republicans on Monday introduced a bill that would avoid a government shutdown at the end of March but that also could mitigate some of the most striking effects of the across-the-board federal spending cuts enacted last week.

But even though the proposed shifts would make the sequester slightly less indiscriminate – particularly for the military – the measure would leave in place the $85 billion spending reduction, locking in the cuts through Sept. 30, end of the fiscal year.

The funding resolution would, for example, prohibit U.S. Immigration and Customs Enforcement from defunding beds in detention facilities where illegal immigrants are being held.

That proposal came after Republicans were angered when several hundred illegal immigrants were released from holding facilities last week in anticipation of the cuts.

The measure also would provide $2 billion in new funding for embassy security, a response to an attack on a U.S. diplomatic outpost in Benghazi, Libya, on Sept. 11 in which four Americans were killed. And it would provide $129 million to help the FBI avoid reducing its staff because of sequestration.

Republicans propose to offset that money with cuts elsewhere, such as extending a salary freeze for federal employees and members of Congress. The freeze would nix a 0.5 percent raise for federal workers that is scheduled to take effect in April under an executive order from President Obama.

These relatively modest changes amount to nibbling around the edges of the new austerity plan introduced with the sequester.

For much of the government, however, the resolution would largely keep spending priorities that had been in place for the first half of the fiscal year — and then whack them by $85 billion, as Obama ordered on Friday.

The proposal is likely to face a vote in the Republican-led House on Thursday. If it passes, it will be up to Obama and Senate Democrats to decide whether to accept its terms: That would mean realizing that reductions they have said could devastate government services are here to stay.

On Monday, Department of Homeland Security Secretary Janet Napolitano set off a flurry of speculation that the administration was exaggerating the impact of sequestration after saying that lines at some of the nation’s busiest airports are “150 to 200 percent as long as we would normally expect.”

The secretary mentioned airports in Los Angeles and Chicago in her remarks at a breakfast sponsored by Politico as examples, although she hedged her remarks saying she would have to check to be sure.

By afternoon, officials with Customs and Border Protection issued a statement saying that wait times had increased significantly at John F. Kennedy and Miami International airports on Saturday with some passengers waiting three hours or longer to clear customs.

Senate Democrats could seek to counter the Republican move by allowing additional flexibility to the mandated cuts in domestic programs, similar to what the House bill would extend to defense cuts.

Senate Appropriations Committee Chairman Barbara Mikulski, D-Md., said Monday that she was still reviewing the House proposal but that she is optimistic that she can work with her House counterpart – Rep. Harold Rogers, R-Ky. – on a speedy agreement for the funding bill.

“He’s sensible, civil and pragmatic,” she said.

Obama has signaled that he wants to avoid a government shutdown when the current funding expires March 27 and that he probably would accept a continuation plan that would allow the sequester cuts to remain in effect for the rest of the fiscal year.

Beyond that, he is likely to turn his attention to undoing the sequester as part of a broader deficit-reduction deal that would replace the current cuts with a combination of new tax revenue and reductions to entitlement programs.

Health and retirement programs are the fastest-growing part of the federal budget but they were shielded as part of the deal that created the sequester in the summer of 2011.

A new deadline this summer to once again raise the nation’s debt ceiling could provide pressure to come up with a deal.

Obama told members of his Cabinet on Monday that he is remains interested in the kind of deficit deal that has been the subject of repeated rounds of failed talks between the White House and Congressional leaders.

“I will continue to seek out partners on the other side of the aisle so that we can create the kind of balanced approach of spending cuts, revenues, entitlement reform that everybody knows is the right way to do things,” Obama said.

White House spokesman Jay Carney said Monday that the effort has included phone calls from Obama to Republican rank-and-file senators — what Carney dubbed the “common-sense caucus” — vigorous Congressional outreach that has been rare for this president.

The White House would not release a full list of the calls, but Congressional sources confirmed they included conversations with Sen. Rob Portman, R-Ohio, and Sen. Bob Corker, R-Tenn., who have long expressed interest in a broad deficit reduction deal.

Obama also spoke Monday with moderate Sen. Susan Collins, R-Maine, and Sen. Tom Coburn, R-Okla., and met last week at the White House with Sens. John McCain, R-Ariz., and Lindsey Graham, R-S.C.

As for the $982 billion continuing resolution, its first test will be the House later this week. Republican leaders will have to soothe anxious conservatives who had rallied around a $974 billion outlay as a ceiling for spending for the year and were surprised to see the bill introduced with $8 billion more.

“There will be some of us very frustrated if we’re suddenly creeping billions of dollars back into the bill,” said Rep. Jason Chaffetz of Utah.

House Appropriations committee staffers will explain at a closed-door meeting with Republicans on Tuesday morning that the measure fully implements sequestration but the higher number is a more accurate reflection of its impact across all government programs.

In the House’s proposal, only the Pentagon would receive broad new help to soften the effect of across-the-board cuts.

The resolution would restore no funding to the military, which absorbed half of the sequester reductions.

But it does reflect new priorities for military spending negotiated between Democrats and Republicans in the House and Senate late last year.

For instance, the House measure proposes adding $10.4 billion to the Pentagon’s operations and maintenance budget.

The money would be offset with other defense cuts.