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“I think we should in a bipartisan fashion stop sequestration before — in the words of the secretary of defense — it destroys the Pentagon,” Mr. Graham said. “But we can cut $1.2 trillion over the next decade without destroying the Defense Department if we choose.”

Mr. Graham suggested putting Mr. Obama’s health care overhaul — his signature domestic achievement — on the table.

“If you want to look at ways to find $1.2 trillion in savings over the next decade, let’s look at Obamacare,” he said. “Let’s don’t destroy the military and just cut blindly across the board.”

Sen. John McCain, Arizona Republican, said he would be open to the idea of closing tax loopholes to raise revenue, but that it is still incumbent on Mr. Obama to step forward and lead on the issue.

“Republicans deserve blame; I’ll take some blame for it,” Mr. McCain said on NBC’s “Meet the Press.” “But the president of the United States is supposed to lead. Why doesn’t he call people over, and we sit down and prevent the sequestration from happening and fulfill the comment he made during the campaign?”