Two days after the 2020 election, Tucker Carlson was furious.

Fox News viewers were abandoning the network for Newsmax and One America News, two conservative rivals, after Fox declared that Joseph R. Biden Jr. won Arizona, a crucial swing state.

In a text message with his producer, Alex Pfeiffer, Mr. Carlson appeared livid that viewers were turning against the network. The message was among those released last week as part of a lawsuit by Dominion Voting Systems against Fox. Dominion, an elections technology company, has sued Fox News for defamation.

At the same time, Mr. Carlson and his broadcasting colleagues expressed grave doubts about an unfounded narrative rapidly gaining momentum among their core audience: that the 2020 presidential election was stolen by Democrats through widespread voter fraud. The belief was promoted by then-President Trump and a coalition of lawyers, lawmakers and influencers, though they produced no evidence to support their assertions.

Many hosts, producers and executives privately expressed skepticism about those claims, even as they gave them significant airtime, according to private messages revealed last week by Dominion. What they said in those messages often differed significantly from what Fox hosts said in public, though they weren’t always contradictory.

Two days after the election, Mr. Pfeiffer said that voices on the right were “reckless demagogues,” according to a text message. Mr. Carlson replied that his show was “not going to follow them.”

But he did follow them. The same day, on “Tucker Carlson Tonight,” Mr. Carlson expressed some doubts about the voter fraud assertions before insisting that at least some of the claims were “credible.”

In the days and weeks that followed, Mr. Carlson was one of several Fox News hosts who repeatedly took a different tone when speaking to viewers on air than when they were talking privately.

The private conversations pose a serious legal threat to the nation’s most-watched cable news network. Dominion has obtained thousands of emails and text messages from Fox employees as part of its $1.6 billion suit. The messages, taken as a whole, are at the core of Dominion’s case.

Fox News has argued in court that the First Amendment protects its right to broadcast false claims if they are inherently newsworthy — and in this case that there was nothing more newsworthy at the time than a sitting president’s allegations of widespread voter fraud.

In a statement, the company said that “the core of this case remains about freedom of the press and freedom of speech, which are fundamental rights afforded by the Constitution” and protected by legal precedent. It added, “Dominion has mischaracterized the record, cherry-picked quotes stripped of key context, and spilled considerable ink on facts that are irrelevant under black-letter principles of defamation law.”

But if a jury looks at the messages from Fox hosts, guests and executives and concludes that people inside the network knew what they were putting on the air was false, it could find Fox liable and reward Dominion with substantial financial damages.

On Nov. 7, 2020, Mr. Carlson told Mr. Pfeiffer that claims about manipulated software were “absurd.” Mr. Pfeiffer replied later that there was not enough evidence of fraud to swing the election.

But during his broadcast on Nov. 9, Mr. Carlson devoted time to various theories, suggesting there could be merit to claims about software manipulation. “We don’t know, we have to find out,” he said.

Mr. Carlson also privately criticized Sidney Powell, a lawyer and conspiracy theorist who was gaining traction among the far right for her involvement in several lawsuits aimed at challenging the election results, the court filings show. Lou Dobbs and Maria Bartiromo, two hosts on Fox Business, a sister channel to Fox News that is also part of Dominion’s lawsuit, repeatedly invited Ms. Powell onto their shows as an expert on voter fraud claims.

Mr. Pfeiffer told Mr. Carlson over text message that election fraud claims, like those being made by Ms. Powell, “need to be backed up.” He warned that President Biden faced being undermined if he was eventually inaugurated.

Mr. Carlson agreed, the filings show.

The next day, Mr. Carlson eviscerated Ms. Powell in a brutal 10-minute monologue, dissecting her claims as unreliable and unproven. He said the show had repeatedly asked her for evidence and, “when we kept pressing, she got angry and told us to stop contacting her.”

In the same monologue, however, Mr. Carlson also gave some credence to Ms. Powell’s claims, saying that “we don’t dismiss anything anymore” and that he is “hopeful” she will come forward with evidence.

Viewers expressed outrage at Mr. Carlson for challenging a prominent Trump ally. And Mr. Trump’s associates quickly jumped to her defense.

Privately, Mr. Carlson continued to criticize Ms. Powell, calling her claims “shockingly reckless.” Mr. Pfeiffer and Mr. Carlson both privately called her a “nut.” Laura Ingraham, who is the host of a 10 p.m. show, and Raj Shah, a senior vice president at the Fox Corporation, the network’s corporate parent, were equally incredulous.

The next day, Mr. Carlson appeared to soften his public stance, suggesting that some of the criticisms about voting machines had merit and concluding, “This is a real issue no matter who raises it.”

Mr. Carlson was far from alone in speaking about Ms. Powell in a different way in private than on the air.

Internally, anchors like Bret Baier appeared surprised to find Ms. Powell getting significant airtime on shows by Ms. Bartiromo and Mr. Dobbs, the court filings show. On Nov. 6, 2020, after someone forwarded Mr. Dobbs’s interview with Ms. Powell, Mr. Baier replied:

The private messages also showed that Ms. Powell was in direct communication with Ms. Bartiromo and Mr. Dobbs, and that she revealed one of the sources for her outrageous claims. The court filings showed that Ms. Powell forwarded an email about voter fraud to Ms. Bartiromo from the source, a woman who claimed, among other things, that “the Wind tells me I’m a ghost.”

If Ms. Bartiromo was deterred by the unusual email, it was not evident to Fox News viewers. Ms. Powell was interviewed on the show the next day.

Consternation over Ms. Powell grew behind the scenes at Fox News as her lawsuits were repeatedly dismissed by courts and her promises to produce concrete evidence of widespread voter fraud never materialized. Yet she was still getting airtime, and senior executives at the network appeared concerned.

Gary Schreier, a senior vice president of programming at Fox Business, said in a private message to Lauren Petterson, the president of Fox Business, that Ms. Bartiromo “has GOP conspiracy theorists in her ear and they use her for their message sometimes.”

Days later, Mr. Schreier received an email from Dominion Voting Systems containing links that refuted Ms. Powell’s voter fraud claims.

That night, Mr. Dobbs interviewed Ms. Powell about Dominion’s comments. But he also used the interview to reinforce her claims of fraud. Mr. Dobbs concluded that “this looks like the effort to carry out an endgame” against Mr. Trump. Ms. Bartiromo interviewed Ms. Powell again two days later.

Several Fox News hosts and producers were criticizing Ms. Powell, including John Fawcett, a producer on Mr. Dobbs’s show, who said he believed Ms. Powell was “doing LSD and cocaine and heroin and shrooms.”

But those criticisms never made it to air. Instead, when Ms. Powell appeared again on Mr. Dobbs’s show days later, she was hailed as a “great American” and “one of the country’s leading appellate attorneys.”

By late November, Mr. Fawcett became increasingly critical of Ms. Powell, according to the court filings. He concluded that she was not verifying her claims. On Nov. 27, 2020, he wrote that her lawsuits were “complete bs.”

Mr. Fawcett also told Mr. Dobbs that Mr. Trump’s legal team had disavowed her. Mr. Dobbs replied that he didn’t know what Ms. Powell was “thinking or doing, Or why!”

But over the next several days, Ms. Powell was invited back by Mr. Dobbs, who echoed her claims that “electoral fraud” was perpetrated by electronic voting machines, “prominently Dominion.”

The next month, after Smartmatic, a competitor of Dominion Voting Systems, sent a letter to Fox News signaling that litigation was imminent, the network put together a video package of an election expert debunking the conspiracy theories that suggested the company’s technology allowed the presidential vote to be rigged. It aired on the programs hosted by Mr. Dobbs, Ms. Bartiromo and Jeanine Pirro.

On Feb. 5, 2021, one day after Smartmatic filed a defamation lawsuit against Fox, Fox Business canceled “Lou Dobbs Tonight.” At the time, Fox said it regularly reviewed its lineup. “Plans have been in place to launch new formats as appropriate postelection, including on Fox Business,” the network said.