Donald J. Trump’s rise has created deep rifts at some of the country’s most powerful political institutions. The latest casualty: Fox News.

In an extraordinary public display of rancor between top-tier news personalities, the Fox anchor Sean Hannity on Wednesday accused his colleague Megyn Kelly of bias toward Hillary Clinton, writing in a late-night tweet, “Clearly you support her.”

Mr. Hannity’s remark came shortly after Ms. Kelly, during her broadcast that evening, pointed out that Mr. Trump had restricted his media appearances in recent weeks to friendly questioners, including Mr. Hannity, who is an overt supporter of the Republican nominee.

“With all due respect to my friend at 10 o’clock, he will go on Hannity and pretty much only Hannity and will not venture out to the unsafe spaces these days,” Ms. Kelly said of Mr. Trump.