On the Dec. 10, 2018, episode of “Tucker Carlson Tonight,” Mr. Carlson referred to The Enquirer’s payment as “ransom” and said Ms. McDougal’s behavior “sounds like a classic case of extortion.” He did not name Ms. McDougal, but he did show a picture of her.

Lawyers for Fox News filed a brief in federal court on Thursday afternoon arguing that the case should be heard there, as opposed to state court, because the damages sought are quite likely to exceed state court limits and because the case is between parties in different states. Ms. McDougal lives in Arizona and Fox News’s headquarters is in New York.

First Amendment protections make defamation lawsuits difficult to win, and the cases often are settled or dismissed by a court before trial.

Lyrissa Lidsky, the dean of the University of Missouri School of Law and an expert in defamation law, said the case against Fox News could come down to whether a reasonable viewer would think Mr. Carlson was accusing Ms. McDougal of a crime.

“The central question of the lawsuit is whether Tucker Carlson was making a factual assertion that she committed the crime of extortion, or if he was using extortion loosely as a form of hyperbole,” Professor Lidsky said, adding that a court may take into account Mr. Carlson’s tone, his wording, and how frequently he makes hyperbolic statements.

Because Ms. McDougal would most likely be considered a public figure, she would have to prove not just that Mr. Carlson’s statements were false and defamatory, but also that he acted with “actual malice” — meaning that he knew or should have known that they were false — in order to win her case.

When The Wall Street Journal first reported on The Enquirer’s payment to Ms. McDougal, a spokeswoman for Mr. Trump said his campaign had no knowledge of it. But Mr. Trump discussed paying The Enquirer for the rights to Ms. McDougal’s story with Michael D. Cohen, who was then his lawyer, according to a secret recording made by Mr. Cohen that was released last year.