“Tucker’s wonderful, he’s provocative, he’s original, but — man oh man,” Mr. Rivera said in a phone interview. “There are some things that you say that are more inflammatory and outrageous and uncorroborated. And I worry that — and I’m probably going to get in trouble for this — but I’m wondering how much is done to provoke, rather than illuminate.”

“Messing around with Jan. 6 stuff … ” Mr. Rivera added, pausing briefly. “The record to me is pretty damn clear, that there was a riot that was incited and encouraged and unleashed by Donald Trump.”

Asked if he would urge his Fox News bosses to reconsider airing Mr. Carlson’s special, Mr. Rivera demurred, saying, “I don’t want to go there, that’s not my job.”

But he added of Mr. Carlson: “He’s my colleague. He’s my family. Sometimes you have to speak out about your family.”