It is unclear how much longer Ms. Van Susteren will be at Fox. When Glenn Beck joined the network in January and posted high ratings in the 5 p.m. time slot, some wondered whether his program would be moved to 10 p.m.

Ms. Van Susteren’s contract will end next year, if it is not renewed. Megyn Kelly, a morning co-host who is considered a rising star at Fox, is the leading candidate to replace Ms. Van Susteren at 10 p.m., according to three people with knowledge of the deliberations, which were intended to be private.

Like Ms. Van Susteren, Ms. Kelly has a legal background and Washington reporting experience. Mr. Shine said the depiction of Ms. Kelly as Ms. Van Susteren’s replacement was “100 percent inaccurate.”