It was also another step in Ms. Brazile’s gradual drift away from the Democratic establishment.

Ms. Brazile became interim chairwoman of the party in July 2016. Days before Mr. Trump’s election in November, emails obtained by WikiLeaks revealed that Ms. Brazile, who worked for CNN at the time, had leaked debate questions in advance to her allies on Hillary Clinton’s campaign staff. CNN promptly fired her. Within a year, she antagonized Democrats with a provocative memoir that accused the party of taking steps to favor Mrs. Clinton over Senator Bernie Sanders during the 2016 primary campaign.

Now, Ms. Brazile will appear on the network that her successor, Tom Perez, the Democratic Party chairman since February 2017, recently barred from sponsoring a party debate because of what he deemed an “inappropriate relationship” with Mr. Trump.

In a statement issued on Monday, Ms. Brazile tried to anticipate any objections.

“I know I’m going to get criticized from my friends in the progressive movement for being on Fox News,” she wrote. “My response is that, if we’ve learned anything from the 2016 election, it is that we can’t have a country where we don’t talk to those who disagree with our political views.

“There’s an audience on Fox News that doesn’t hear enough from Democrats,” Ms. Brazile continued. “We have to engage that audience and show Americans of every stripe what we stand for rather than retreat into our ‘safe spaces’ where we simply agree with each other.”

On the air with Ms. Perino on Monday afternoon, Ms. Brazile identified herself as “a progressive and a liberal.” She noted that she had occasionally appeared on Fox News in the past, including as a guest host of the opinionated talk show “The Five.”