Fox News star Megyn Kelly has changed agents and publicity teams since last year. Now the question is if she will change TV networks.

Host of “The Kelly File,” one of the cable-news channel’s most popular shows, Ms. Kelly is in active talks over her contract, which expires next July. Her profile has been rising during the presidential election cycle, in part thanks to a dust-up with Republican candidateDonald Trump.

Keeping Ms. Kelly is a priority for senior management, including Rupert Murdoch, chief executive of Fox News and co-executive chairman of its parent company, 21st Century Fox .

Asked if Ms. Kelly would stay at the channel, Mr. Murdoch said in an interview that she is important to the network and he hopes to get a contract signed “very soon,” but noted, “it’s up to her.”

Mr. Murdoch said he is kept abreast of the talks “every minute of the day.” While he doesn’t want to lose her, he said, “we have a deep bench of talent, many of whom would give their right arm for her spot.”