Fox News viewers “value our product so much, they go to hotels and if they can’t have Fox, they send us emails. They go on cruises, and if they can’t have Fox, they send us emails,” Mr. Finley said. “This is a way for us to meet that demand.”

Whether the venture would be a moneymaker is up in the air.

Fox News reaps more than $1 billion in annual profit, providing ample funds to hire a new team for Fox Nation, which is not expected to initially carry advertising. Mr. Finley declined to estimate his start-up costs, and streaming services in conservative media have had a mixed record of success.

The Blaze, a web-only service founded by the host Glenn Beck in 2011 after he left Fox News, struggled to attract interest and eventually morphed into a more traditional network distributed by cable and satellite providers. Bill O’Reilly, who was fired by Fox News in April, started a subscription service on his website that has earned little attention.

Mr. Finley said Fox Nation was not comparable to a personality-driven product. “This is not starting from scratch here,” he said. “Glenn Beck had a ton of viewers when he was here on Fox. When he left, it didn’t seem to me that they followed him. People are loyal to the Fox brand.”

The median Fox News viewer is 65 years old, according to Nielsen, but the network points to its website traffic and heavy presence on Facebook and other social media platforms as evidence that a web-only service can appeal to its audience.

Among Fox News’s main rivals, MSNBC has no stand-alone product. CNN has a streaming service, CNNgo, which offers some free original programming, but it otherwise requires an existing cable or satellite subscription. Jeff Zucker, CNN’s president, said in December that he was considering a digital product for the channel’s “Great Big Story” brand, which is aimed at younger viewers.