BRUSSELS—The European Union is cautiously betting on its top official, Jean-Claude Juncker, to persuade President Donald Trump not to escalate the trans-Atlantic trade fight.

Appeals from European leaders have failed to dissuade Mr. Trump from imposing tariffs on imports of steel and aluminum and threatening new ones on cars, but officials say they believe Mr. Juncker’s blunt speech and disdain for protocol has won some begrudging respect from the U.S. leader.

During...