WHITE PLAINS, N.Y.—Democratic presidential candidate Hillary Clinton’s top advisers on Saturday sharply criticized the Federal Bureau of Investigation’s review of newly discovered emails and demanded Director James Comey provide the public with additional information to explain the new probe.

Calling the FBI’s actions “extraordinary,” Clinton campaign Chairman John Podesta said the FBI director’s letter announcing the review was “long on innuendo and short on facts.” Mr. Podesta blasted the FBI’s actions, saying Mr. Comey...