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Republicans argue that while they don’t like some of the specific cuts, particularly to defense, the party will not accept more taxes and wants a different set of spending cuts that would cover entitlement spending.

“Personally, I don’t believe the world will end if the president’s sequester takes effect,” said Senate Minority Leader Mitch McConnell, Kentucky Republican. “But our country would be much better served if the Democrats who run Washington would get off the campaign trail and work with us to trim the budget in a more rational way.”

The Senate will hold votes this week on two proposals: a Democratic bill to cut agriculture subsidies and impose a new minimum tax rate on the wealthy in order to cancel half of the sequesters, and a Republican bill that likely would try to give the administration the ability to move money around to blunt the worst cuts.

In the meantime, though, the White House has highlighted some of those worst-case scenarios.

Homeland Security Secretary Janet A. Napolitano on Monday told reporters at the White House that she would be forced to furlough Border Patrol agents, pulling them from their rounds along the U.S.-Mexico border.

She also hinted at the decision to release illegal immigrants, saying she would not be able to maintain the full slate of 34,000 detention beds mandated by Congress.

“How am I supposed to pay for those? There’s only so much I can do,” she said.

United We Dream, an immigrant rights group, said the releases show that the administration had been keeping folks detained who never should have been there.

“Low-priority individuals — people who pose absolutely no risk or danger to society, but rather are upstanding members of their communities and families — should not have been locked up to begin with,” said Carolina Canizales, coordinator of United We Dream’s End Our Pain program.

Republicans said the homeland security budget remains high and that the total amount of cuts is only slightly more than 5 percent of ICE’s budget. They argued that the savings could come from maintenance funds instead of operations.

House Judiciary Committee Chairman Bob Goodlatte, a Virginia Republican who oversees ICE, called the decision to release immigrants back into society “abhorrent.”

“By releasing criminal immigrants onto the streets, the administration is needlessly endangering American lives,” the Virginia Republican said in a statement. “It also undermines our efforts to come together with the administration and reform our nation’s immigration laws. Unfortunately, this Administration has a poor record of enforcing our immigration laws and has routinely sought to undermine them.”

Ms. Napolitano on Monday also warned about an increase in illegal immigrants if the sequesters take effect, arguing that the cuts will mean fewer agents to patrol the border or watch the ports of entry.

Illegal entries had been declining in recent years, though a rise in border arrests last year suggested that the trend may have halted.

Officials also have expressed concern about spillover violence from the brutal drug war in Mexico, but a study released Tuesday by the Government Accountability Office found that crime rates in the counties along the U.S.-Mexico border are down.

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