For more than six decades, media titan Rupert Murdoch mostly was a buyer, rather than a seller. Not anymore.

In October, Walt Disney Co. Chief Executive Robert Iger phoned Mr. Murdoch, the executive chairman of 21st Century Fox, to propose a purchase that would hive off Fox assets worth some $40 billion. Changes in the media landscape had altered Mr. Murdoch’s thinking. A deal could also clarify the family’s succession plans.

21st...