LOS ANGELES — In late summer, Verizon Communications came to Rupert Murdoch with a surprise acquisition offer.

Verizon — locked in battle with AT&T, which was then finalizing its $85.4 billion takeover of Time Warner — wanted to buy pieces of 21st Century Fox, Mr. Murdoch’s television and film conglomerate. Representatives of the two companies secretly met at least once to discuss a merger.

Mr. Murdoch, 86, shrugged off the talks as uninspiring, according to an associate, who spoke on the condition of anonymity because he wanted to maintain his access to the media titan. Verizon declined to comment, but the overture prompted Mr. Murdoch to start to think seriously — for the first time — about selling his Hollywood treasures.

Not only would a sale solve a business problem, it could solve a family one.

Several months later, Mr. Murdoch agreed to sell much of 21st Century Fox to the Walt Disney Company. The proposed $52.4 billion deal, which is subject to regulatory approval, has the potential to radically reshape the entertainment world, but it also has many wondering what the future holds for Mr. Murdoch and the two sons who seemed to be on the cusp of taking over his vast media holdings.