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.”

Mr. Murdoch and his family are major stakeholders in both 21st Century Fox and Wall Street Journal owner News Corp.

For the final year of her current deal, Ms. Kelly is set to receive about $15 million, people familiar with the matter said. The people said she is seeking an average annual salary north of $20 million for her next contract—which would put her on par with Fox host Bill O’Reilly…

The networks seen as most likely to make an aggressive play for Ms. Kelly are CNN, where she could go up against Fox News in prime-time, and ABC, where TV executives say she might fit well on “Good Morning America” or in an evening newsmagazine. CNN and ABC declined to comment.