A graduate of Harvard Law School, Ms. Wiehl started making regular appearances on Mr. O’Reilly’s show in 2001, when she joined Fox News as a legal analyst. During a segment in September of that year, Mr. O’Reilly announced that Ms. Wiehl had landed a job at the network and said she owed him.

“Hey, you know, Lis, I got you this job,” he said. “You know that?”

“I know you did, I know,” she replied.

“So you owe me,” Mr. O’Reilly said. “You owe me big.”

“No, no, no,” Ms. Wiehl said.

Mr. O’Reilly also made suggestive remarks to Ms. Wiehl on the air. During one segment on his radio show in 2005 about a strip club, he suggested that she learn how to dance for a $10,000 tip.

Ms. Wiehl last appeared on Mr. O’Reilly’s show on Dec. 20, 2016. On Jan. 2, Mr. O’Reilly received a draft of a lawsuit Ms. Wiehl was threatening to file outlining her allegations of sexual harassment, and 21st Century Fox received a copy of the complaint soon afterward.

Both Mr. O’Reilly and 21st Century Fox were at critical junctures. If the allegations became public, they would not only embarrass Mr. O’Reilly and harm his career, but could jeopardize his yearslong custody battle with his ex-wife. A hearing was set for later that month, when Mr. O’Reilly’s lawyers planned to argue that he should be given more time with his son, according to two people familiar with the dispute.

At Fox News, Ms. Kelly had just announced that she was leaving the network for NBC. Her departure made Mr. O’Reilly’s presence in the prime-time lineup even more crucial, with his show pulling in top ratings and generating hundreds of millions of dollars in revenue.

Mr. Newman told 21st Century Fox that Mr. O’Reilly considered it a personal matter and that he would resolve it on his own. Mr. Newman handled the negotiations with lawyers for Ms. Wiehl.

After a few days of negotiation, Mr. O’Reilly and Ms. Wiehl reached a deal, according to a copy of the term sheet for the agreement that was sent anonymously to The Times and confirmed by the people briefed on the settlement. Dated Jan. 7, it called for Ms. Wiehl to be paid over a period of time to ensure her silence. In return, she agreed not to sue Mr. O’Reilly, Fox News or 21st Century Fox. And all photos, text messages and other communications between the two would be destroyed.

Ms. Wiehl signed an affidavit, dated Jan. 17 and obtained by The Times, stating that the two sides had resolved their dispute and that she had “no claims against Bill O’Reilly concerning any of those emails or any of the allegations in the draft complaint.” In the affidavit, she said she had worked as a lawyer for Mr. O’Reilly and was serving in that capacity when he sent her “explicit emails that were sent to him.”

In response to questions about why he sent sexually explicit material to Ms. Wiehl, Mr. O’Reilly said that during his time at the network, he had been sent threatening messages almost every day, including some that had obscene material. To deal with this problem, Mr. O’Reilly said, he set up a system in which the material would be forwarded to his lawyers so they could evaluate whether he needed to take any legal action. Mr. O’Reilly said Ms. Wiehl was among those lawyers.

Although the matter had been settled confidentially, Mr. O’Reilly’s lawyers were concerned about keeping the dollar figure secret. Mr. Newman provided the company with a document that informed them of the deal but did not include the dollar figure.

The company said Mr. Newman made clear that it would not be told the financial terms because Mr. O’Reilly thought the company “leaked sensitive information.”

In February, Mr. O’Reilly received his new contract, with a salary increase to $25 million, from about $18 million. It’s not clear who initiated negotiations for the extension. Mr. Newman says Fox News pushed to renew the contract; the company says the negotiations were bilateral.

“It was Fox News that wanted to renew Bill O’Reilly because of the Megyn Kelly defection,” Mr. Newman said, adding that Mr. O’Reilly was a wealthy man who had no need for extra money. The company said it would have renewed his contract whether Ms. Kelly stayed or left.

Mr. Fabiani, Mr. O’Reilly’s representative, said that he was concerned that 21st Century Fox’s statements about Mr. O’Reilly were designed to hurt his brand.

“Up to this point, Fox News and Mr. O’Reilly have had a constructive business relationship — with Fox News even running ads for his new book on their air,” Mr. Newman said. “We hope that all the leaks coming out of Fox are not designed to hurt Bill O’Reilly in the marketplace.”

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