WASHINGTON—Late Sunday, a group representing thousands of FBI agents told its members it was optimistic about the agency’s future, in part because its new director, Kash Patel, had assured the group he would choose a deputy from among their ranks.

FBI deputy director Dan Bongino is a bombastic podcast host who has never worked at the FBI and has spent years pushing conspiracy theories about the bureau.

President Donald Trump announced on Sunday that Dan Bongino, a conservative talk show host, will be the next deputy director of the Federal Bureau of Investigation.