There is an unlikely new hot spot for Democratic candidates: Fox News.

President Trump’s favorite network is increasingly playing host to hopefuls from the Democratic presidential field, eager for exposure to the vast Fox News audience — even as they risk a backlash from others in the party who view the network as an ideological menace.

The expedition into what many liberals consider enemy territory picked up this week after Senator Bernie Sanders of Vermont appeared at a town hall on the network, drawing the biggest television audience of any 2020 Democratic candidate so far — more than 2.5 million people — while pitching himself to Trump-leaning viewers who may be willing to cross party lines next year.

On Wednesday, Senator Amy Klobuchar of Minnesota said she had agreed to a Fox News town hall-style event next month. Mayor Pete Buttigieg of South Bend, Ind., is in advanced talks with the network. Julián Castro, the former housing secretary, is close to signing on, and Senators Kirsten Gillibrand of New York and Cory Booker of New Jersey say they are open to the idea.

The debate over whether to appear on Fox News reflects in some ways a larger divide in the party as it ponders how to retake the White House: Should Democrats focus on expanding and mobilizing the various coalitions that make up their base, or seek inroads with the millions of Americans who supported Mr. Trump in 2016?