WASHINGTON—A proposal under consideration at the White House to reverse decades of U.S. nuclear policy by declaring a “No First Use” protocol for nuclear weapons has run into opposition from top cabinet officials and U.S. allies.

The opposition, from Secretary of State John Kerry, Secretary of Defense Ash Carter and Secretary of Energy Ernest Moniz, as well as allies in Europe and Asia, leaves President Barack Obama with few ambitious options to enhance his nuclear disarmament agenda before leaving office, unless he wants...