BUDAPEST—Hungarian Prime Minister Viktor Orban has told U.S. diplomats that he wants his country to be “neutral, like Austria” as Washington pushes for a tougher line on Russia and China, deepening fears that a longtime American ally is drifting from its orbit.

Aides to Mr. Orban said he wants to remain a troop-contributing member of the North Atlantic Treaty Organization and values the security Hungary gets from the alliance. But a senior Hungarian official said Mr. Orban strongly objects to U.S. pressure aimed at curbing...