Rupert Murdoch is facing a familial battle royale as three of the 93-year-old media mogul’s four adult children fight his efforts to adjust the family trust in order to leave eldest son Lachlan Murdoch in charge after his death, according to an investigative report published Wednesday by the New York Times.

The Times report is based in part on a 48-page decision by a Nevada probate commissioner that was filed under seal but obtained by the newspaper. Three of Rupert Murdoch’s four adult children — Prudence, Elisabeth and James — are fighting their father’s efforts to adjust the trust. The battle has so far played out in secret but will now be closely watched as the probate commissioner considers petitions filed by the three younger Murdochs.

According to the Times, Rupert Murdoch is motivated by the desire to ensure that Fox News, the New York Post, the Wall Street...