Walt Disney Co. is close to a deal to acquire a large piece of 21st Century Fox Inc., people familiar with the situation say, in a pact that could help the entertainment giant accelerate its ambitions in streaming media, shore up its television business and grab hold of lucrative movie franchises.

The deal, expected to be announced Thursday, would value the assets Disney is acquiring at $60 billion, including debt. Those assets include the Twentieth Century Fox movie and TV studio, cable channels including regional sports...