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“Instead of using our military men and women as campaign props, if the president was serious, he’d sit down with [Senate Majority Leader] Harry Reid and begin to address our problems,” House Speaker John A. Boehner told reporters in an afternoon news conference.

Sen. Tom Coburn, Oklahoma Republican and Congress’ top waste-watcher, said the federal government can begin to control spending immediately.

He sent a letter to the White House budget office Monday pointing to job openings the government has advertised, including 10 slots for drivers at the State Department, at wages up to $26.45 an hour, and a Labor Department staff assistant to handle scheduling and screen phone calls, at a salary of up to $81,204 a year.

“Are any of these positions more important than an air traffic controller, a border patrol officer, a food inspector, a TSA screener, or a civilian supporting our men and women in combat in Afghanistan?” Mr. Coburn asked in his letter to White House budget office acting Director Jeffrey Zients.

The budget office didn’t respond to a request for comment Monday.

Slow rollout

The sequester cuts won’t be felt immediately, as agencies decide how to impose the cuts over the rest of this fiscal year, which ends Sept. 30.

Congress will get a do-over in late March when it reaches a deadline for a set of annual appropriations bills to keep the government open — giving lawmakers a chance to rewrite all discretionary spending levels.

For now, though, the political battle remains intense.

As part of its public relations war, the White House has been bringing out Cabinet members such as Ms. Napolitano and Transportation Secretary Ray LaHood to warn that the cutbacks will result in longer waits at airports and ports of entry as a result of furloughed federal employees.

Ms. Napolitano said the cuts to her department will endanger the country by hurting border security and cybersecurity efforts.

“There’s always a threat,” she told reporters. “We’re going to do everything we can to minimize that risk. But the sequester makes it awfully, awfully tough.”

In addition to fewer agents on the border, she said she likely would have to cut the number of detention beds to hold illegal immigrants.

But Mr. Coburn, in a letter to Ms. Napolitano sent Monday, argued that she has flexibility to decide which cuts go into effect and said her department is poised to carry over $9 billion in unspent money at the end of this year — “raising the question of why we would not start by reclaiming these funds.”