Federal prosecutors are looking into whether Fox News Channel and its parent company tried to disguise a $3.15 million payment to a former employee who said she had a 20-year affair with the network’s former chairman, Roger Ailes, according to people involved with the investigation.

Investigators in the U.S. Attorney’s Office for the Southern District of New York have focused on a payment to Laurie Luhn, a former Fox booker and event planner who left the company in 2011 with the seven-figure severance package. Luhn later claimed that she had engaged in a consensual but a mentally abusive, relationship with Ailes and that several of his lieutenants facilitated the assignations and were aware of his alleged mistreatment of her.

Prosecutors have taken testimony from several witnesses, including Luhn, about her severance package and how it was recorded on Fox’s books, according to those familiar with the probe.

The size of the payout is of less concern to the investigation than the manner in which it was accounted for, those people said. Prosecutors are investigating whether Fox News Channel and its parent company, 21st Century Fox, improperly accounted for the payments to Luhn and other ex-employees to minimize their impact on Fox’s books.

In addition to Luhn, they have taken testimony from at least two other former Fox employees, including former chief financial officer Mark Kranz and former Fox media relations chief Brian Lewis, over the past six weeks, according to people involved with the investigation.

Kranz, who retired from Fox in August, is seen as a potentially crucial witness for the prosecution; people close to the investigation have indicated that he has been offered immunity. Fox reportedly paid $8 million in 2013 to settle Lewis’s contract.

Luhn’s case is of particular interest to prosecutors because of the unusual nature of her severance. One of her settlement checks is signed by a corporate executive based in Los Angeles who had no direct involvement with Fox News, according to people who spoke about the investigation anonymously because they weren’t authorized to speak on the record.

Luhn worked primarily at Fox’s headquarters in New York, not in Los Angeles, so the executive’s signature has raised questions about where the funds for her severance came from. Investigators have been looking into whether some of the cost of her settlement was attributed to another Fox-owned entity to minimize the impact on Fox News’ financial statements.

Luhn, who signed a nondisclosure agreement and was not available for comment, nevertheless told New York magazine last summer about her two-decade affair with Ailes. Luhn told the magazine that the stress of the relationship led to a series of mental breakdowns. She claimed that other Fox executives helped cover up the relationship. Ailes has previously denied any sort of improper relationship with Luhn.

A second issue in the investigation concerns the structuring of other settlements to former employees. Fox News settled at least one such claim involving a former high-ranking manager by stretching the payments over several years, in effect accounting for the executive’s severance compensation as if he were still an employee instead of taking a one-time financial hit.

People familiar with the investigation said it was still in an early fact-gathering phase. There have been no indications about when it could conclude or whether criminal charges would result; 21st Century Fox has previously said that it was cooperating with the probe.

“Mr. Ailes took no part in any decisions about the accounting, reporting or disclosing of any severance to Ms. Luhn,” said Susan Estrich, Ailes’s lawyer.

Fox News referred questions to 21st Century Fox, whose spokesman, Nathaniel Brown, declined to comment.

One of the attorneys handling the probe for the prosecution, Damian Williams, declined to comment. A second federal prosecutor who is involved, Andrea Griswold, did not respond to a request for comment.

Although the payments to Luhn and other employees are relatively small, particularly within a multibillion-dollar company such as Fox, the heart of the matter for prosecutors is whether they constituted “material” events requiring public disclosure to investors under federal securities law.

A series of such payments could be considered material not because of their size but because they could raise concerns among investors about the stability of the company’s management or finances.

As the co-founder and chief architect of Fox News, Ailes was a towering figure within the network. Any sense that his reign was threatened by multiple allegations of harassment could constitute a material event. The same could be true of O’Reilly, the network’s biggest star and a chief moneymaker.

Ailes was pushed out in July after former Fox host Gretchen Carlson filed a sexual-harassment lawsuit against him, triggering a series of harassment complaints against Ailes by women who had worked for him. Carlson later settled with Fox for $20 million.

O’Reilly was fired last month following a New York Times investigation that found he and Fox had paid $13 million to settle five harassment complaints by female employees since 2002. Both O’Reilly and Ailes have denied the claims against them.

The parent company has disclosed only the costs associated with the Carlson lawsuit. In an investor disclosure document filed Nov. 2, it said that it spent “approximately $35 million . . . related to settlement of pending and potential litigations” in the wake of Ailes’s resignation. The money was divided among Carlson, Fox’s attorneys and an unknown number of other employees whom Fox paid to address the complaints.

Correction: An earlier version of this article incorrectly said former Fox host Juliet Huddy had been questioned by prosecutors in the probe. Her attorney said she has not done so.