Back on the air Wednesday night, Mr. Carlson gleefully extended the feud, mocking Mr. Smith’s comments and noting, “Unlike maybe some dayside hosts, I’m not very partisan.” (Mr. Smith, a nonpartisan news anchor, is often accused by Mr. Trump of a liberal bias.)

The on-air bickering came as Fox News journalists embarked on a sensitive task: covering the details of Mr. Trump’s interactions with Ukraine even as their network’s star commentators have striven to dismiss the matter as a “witch hunt” led by Democrats.

On Friday, Chris Wallace, the “Fox News Sunday” anchor, observed that the spin from Mr. Trump’s defenders “is not surprising, but it is astonishing, and I think deeply misleading.”

Mr. Wallace did not specify which defenders he was referring to. But liberals were quick to point out that Fox News’s prime-time hosts have been among Mr. Trump’s most vocal allies in the past few days. On Thursday, Sean Hannity opened his Fox program by addressing, in his words, “the radical, destructive, delusional Democratic Party and the media mob, their allies in crime, which has now descended into complete and utter madness.” (Fox News said that Mr. Wallace was not referring to his prime-time colleagues.)

Friction between Fox News’s news reporting staff and its pro-Trump lineup of prime time commentators is not new. Journalists in the network’s Washington bureau have grumbled about the president granting interviews to pundits like Mr. Hannity rather than the White House team.