Mr. Hannity and other anchors, including the morning show hosts on “Fox & Friends,” rarely question the president, and several of them recently dined with Mr. Trump at the White House. Rupert Murdoch, Fox News’s executive chairman, is a regular adviser to Mr. Trump, even counseling him on the fate of Stephen K. Bannon, the president’s chief strategist.

But Mr. Trump has his critics on Fox News. Juan Williams regularly battles with his pro-Trump co-hosts on the prime-time show “The Five.” The daytime news anchor Shepard Smith can offer up tough criticism. “So far we’ve been unable to find the very fine people protesting with the white supremacists,” he said sarcastically on Wednesday’s broadcast, referring to Mr. Trump’s assertion that “very fine people” were among the demonstrators in Charlottesville.

Ms. Williams’s criticism was more scathing. Addressing Mr. Trump directly on the air Monday, she said: “While you personally may not be a racist, President Trump, what you are is all too happy to reap the benefits of their support, and you even tacitly encourage them with evasive, irresponsible statements.”

Ms. Williams, a lawyer who formerly worked at CBS News, said in the interview Wednesday that her superiors at Fox News had not expressed concern.

“I’ll tell you the truth, if they had said anything, it would have shocked me,” she said. “I’m no puppet for the network. I call Fox as I see Fox. And for me it’s been a positive experience. It’s not been a perfect experience, but I’m there because the good outweighs the bad.”

Her “Specialists” co-host, the libertarian commentator Kat Timpf, has also drawn attention for criticizing Mr. Trump on the air, calling his news conference “one of the biggest messes that I’ve ever seen.”