In August, Congressional Republicans tried to box in Democrats and the White House by demanding huge deficit cuts in exchange for preventing a government default. Then they joined in the creation of a “supercommittee” on deficit reduction that they hoped would take taxes off the table and focus entirely on cuts in spending.

But that supposed victory has forced many Republicans into an equally tight corner. They are starting to realize that if they remain adamant, the resulting across-the-board cuts will disproportionately affect programs they support, starting with military spending.

The joint committee created by the debt-ceiling agreement is desperately groping behind closed doors for ways to cut at least $1.2 trillion from the federal deficit. Republican leaders want it all to come from spending cuts; Democratic leaders want a mix of cuts and revenue increases. If the two sides cannot agree, there will be automatic cuts, which largely spare social-welfare programs but would severely reduce military and security spending.

Senator Jon Kyl of Arizona, who is happy to cut virtually anything else, says military cuts could lead to job losses. Representative Howard McKeon of California, the Armed Services Committee chairman, is so rattled by the law that he used an old scare tactic, saying the cuts could produce a new military draft. “It is my suspicion that the White House and Congressional Democrats insisted on that defense number for one purpose: to force Republicans to choose between raising taxes or gutting defense,” Mr. McKeon said.