WASHINGTON—The U.S. government moved to within hours of its first shutdown since 1996, as House Republicans redoubled their drive early Sunday to delay the new health care law and Senate Democrats stood firm against changing the law as a condition of funding federal departments.

The standoff left little prospect that Congress could reach agreement on terms for funding the government by midnight Monday, when the current fiscal year expires. A shutdown would leave essential services operating but prompt federal agencies to suspend...