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.

There will still be an institution called the Federal Bureau of Investigation, but with Kash Patel and Dan Bongino at the helm, it will not be the FBI.

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.