Until Megyn Kelly, no prime-time Fox News anchor had attempted the leap from partisan basic cable to the more pedigreed world of network news.

Less than a month into her tenure at NBC, Ms. Kelly and her new employer — which has placed a multimillion-dollar bet on her success — are learning just how daunting the transition can be.

Ms. Kelly’s coming interview with Alex Jones, the conspiracy-monger and influential voice of the alt-right, has generated a fierce backlash from critics on the right and the left, just as the anchor is trying to introduce herself to a broader national audience.

Parents of children killed in the 2012 Sandy Hook elementary school shooting, which Mr. Jones has called a hoax, asked NBC to spike the interview, saying it was hurtful to those who had suffered such a grievous loss to offer a platform for Mr. Jones’s views. Ms. Kelly was disinvited from a Sandy Hook charity event and accused by some viewers of chasing ratings by infecting NBC with Fox News-style conservatism.

The latest setback came early Friday morning, when Mr. Jones’s website, InfoWars, published audio of Ms. Kelly cajoling and flattering her interview subject as she tried to secure his cooperation for the segment. “I’m not looking to portray you as some boogeyman,” Ms. Kelly can be heard saying.

Assurances of fair coverage are standard practice in television journalism, where anchors seeking access routinely present their intentions in the best possible light. So far, NBC is standing by Ms. Kelly, urging viewers to withhold judgment until the segment airs on Sunday night on Ms. Kelly’s newsmagazine show.

But the firestorm has been an unwelcome surprise at the network, which is banking on Ms. Kelly, who is drawing a reported salary of around $15 million, as its next flagship star.

Her new show, on which the Jones interview is set to air, already faced an uphill climb against CBS’s “60 Minutes,” the No. 1 show in television news and a daunting competitor for any anchor. And Ms. Kelly is about three months away from taking over the 9 a.m. hour on the “Today” show, a coveted soft-news time slot where she will be expected to be less Mike Wallace, more Oprah Winfrey.

No morning-TV personality wants to face the wrath of families of victims of a school shooting. And Ms. Kelly, who is predominantly known as the face of a conservative-leaning cable news network, does not have a reservoir of good will with NBC’s bigger audience to fall back on.

“It’s Jimmy Fallon tousling Trump’s hair,” said Martin Kaplan, a professor of media and entertainment at the University of Southern California, likening the Kelly-Jones tempest to the moment last fall that is largely credited as causing lasting damage to Mr. Fallon, NBC’s “Tonight Show” host. “It deprives the moment of the gravity that it has in our society.”

Some critics have urged NBC to quash the segment. But journalists, by and large, have offered support for Ms. Kelly, saying that Mr. Jones, who has spoken to President Trump and offered him advice, is a legitimate subject for journalistic scrutiny. Other shows, like Piers Morgan’s prime-time CNN show and ABC’s “Nightline,” featured interviews with Mr. Jones in 2013. Mr. Jones’s beliefs, however suspect or offensive, reach a sizable audience around the country thanks to his popular radio show and website.

“Megyn is a very good journalist, and I expect, especially in light of everything that’s been said this week, that he will be held to account,” Jeff Zucker, the president of CNN, said during a question-and-answer session with journalists this week.

But Mr. Zucker added that NBC had “done themselves no favors” in the way the segment has been marketed to viewers.

He also cited a damaging photograph of Ms. Kelly wearing sunglasses and smiling alongside Mr. Jones on the day of their interview. The photograph was distributed by Mr. Jones, not NBC, but by posing for it Ms. Kelly conveyed the sense of a cozy encounter rather than a cross-examination.

“If you are going to do this story, the tease needs to be you holding up a picture of the dead kids at Sandy Hook and saying, ‘How dare you?’ And that’s what you need to do,” Mr. Zucker said.

NBC has declined to release a transcript of Ms. Kelly’s interview, leading to speculation that her interrogation of Mr. Jones over his more contentious views — for instance, that Sandy Hook was a hoax and the 9/11 terror attacks were an inside job — was less than ferocious. The New York Post reported that NBC executives were retooling the segment in light of the criticism, and that Ms. Kelly was seeking parents of Sandy Hook victims who would be willing to speak on-camera in response to Mr. Jones.

“We remain committed to giving viewers context and insight into a controversial and polarizing figure, how he relates to the president of the United States and influences others, and to getting this serious story right,” NBC said in a statement on Friday. “Tune in Sunday.”

Still, the storm shows few signs of abating. Late Thursday, a law firm representing several Sandy Hook families sent a letter to NBC warning that “airing the interview will cause serious emotional distress to dozens of Sandy Hook families.” At least one advertiser, JPMorgan Chase, has pulled its sponsorship of Sunday night’s show, and Ms. Kelly admitted this week that she was taken aback by the level of backlash at the Jones segment.

Chris Cuomo, the CNN anchor, said he believed network executives would be toiling until the last minute to ensure that Sunday’s segment is seen as responsive to the public criticism.

“Coming from that world, I do not envy the job those producers and editors have right now,” Mr. Cuomo said, adding a profane joke about all the re-cutting NBC must be doing right now.

Mr. Zucker nodded vigorously at Mr. Cuomo’s remarks. “It’s true,” he said. “Absolutely right.”

Kristin Hussey contributed reporting.