Mr. Mishkin runs Fox News’s “decision desk,” the team responsible for telling Fox viewers — also known as Donald Trump’s base — who won the election. The team and its sister polling unit are among the few endeavors at Fox that have proven immune to the president’s takeover of the network. Mr. Mishkin is a straight shooter — a registered Democrat who told me he voted for Hillary Clinton in 2016, and is paid as a consultant, not as a Fox employee.

“We’re going to come under enormous pressure,” another top Fox figure told me of election night.

The nightmare scenario goes like this: It’s a close race, and Mr. Trump leads in the early vote count in Pennsylvania, and needs just that state to win the election. Tens of thousands of votes are still untallied, and the counting may take weeks — but Mr. Trump has already declared that he’s been re-elected. He’s demanding that Fox do the same, making calls to Fox Corporation’s co-chairman, Rupert Murdoch, or working back channels to the executive who effectively runs the network, Viet Dinh. Mr. Trump’s most loyal acolytes at Fox, the prime-time hosts Sean Hannity and Laura Ingraham, are backing the president’s claim on the air. And Fox faces the temptation it often succumbs to: offering its audience the alternate reality it wants.

“There’s real concern about the choice Fox is going to make given its own history,” said Vanita Gupta, the president and chief executive officer of the Leadership Conference on Civil and Human Rights, which is among the groups pushing journalists to take their time in covering the vote counts. “Are they going to be a state media organ and sabotage our democracy in the process, or are they going to show up as media that is going to be thoughtful and careful about prematurely calling results before all the ballots are counted?”