WASHINGTON—A push by some Senate Democrats to suspend the federal gasoline tax is running into opposition from both Republicans and other Democrats, presenting a challenge to the nascent attempt to reduce rising pump prices that have worried the White House.

Sponsored by several Senate Democrats who face competitive re-election races this fall, the legislation would halt the tax of roughly 18 cents a gallon for the rest of the year. The highest rate of inflation in decades poses a political problem for Democrats trying to hold their control of the 50-50 Senate, and supporters see the measure as a quick way to try to reduce costs faced by many Americans.