WASHINGTON — With President Obama insisting on higher tax rates for affluent Americans and winning public support for the idea, congressional Republicans find themselves in an increasingly difficult political spot and are quietly beginning the difficult search for a way out.

Senior Republican leadership aides say they are contemplating a fallback position since a standoff over expiring tax rates will be portrayed by Democrats as evidence that the opposition is willing to allow taxes to rise on the middle class to keep taxes from rising on the rich — and their intransigence could mean taxes go up on rich, poor, and middle class alike.