Federal authorities have interviewed current and former Fox News employees and on-air talent in a widening inquiry into the nature of sexual-harassment settlements and alleged intimidation tactics at the network, according to people familiar with the probe.

Prosecutors at the U.S. attorney’s office in Manhattan and criminal investigators from the U.S. Postal Inspection Service have spoken to women who accused Roger Ailes, former Fox News chairman and chief executive, of sexual harassment, the people familiar with the probe...