WASHINGTON—Republicans head into the next budget battle with President Barack Obama torn between two long-standing goals: Strengthening the military and cutting federal spending.

The prospect of deep cuts in defense is troubling to many in the party, which has traditionally supported robust defense spending. But increasingly, that impulse is giving way to arguments from GOP lawmakers, many of them new to Congress, who say the most important goal is to rein in federal deficits. They believe that steep, across-the-board spending...