Mr. Iger does have a history of paying premiums for properties he wants, albeit on a much smaller scale. The most obvious example is Pixar, which Disney bought in 2006 for $7.4 billion, or $9.4 billion in today’s money. The price was criticized at the time as exorbitant — wrongly, as it turned out, given the value that Disney has since gotten from the animation studio.

Mr. Iger said on an earnings conference call last month that Disney’s plans to introduce an entertainment-focused streaming service next year were “not dependent at all on the assets we’re buying from Fox.” But the Fox content would certainly help make Disney’s streaming strategy a success.

As part of the deal, Mr. Iger agreed to delay retirement to the end of 2021, from July 2019. Until now, his run as chief executive — he took over in 2005 — has been considered a triumph. Comcast’s stealing Fox would become the defining event of his tenure.

Comcast’s new offer is about 19 percent higher than Disney’s proposal, according to its statement. It also includes contractual assurances such as a reverse breakup fee — worth about $2.5 billion — in the event a transaction is blocked by the government.

Mr. Murdoch and his company’s board had rejected Comcast’s earlier offer partly because of concerns that the government would block the deal, but the AT&T-Time Warner decision allayed many of those worries.

Mr. Roberts needed to move quickly. Fox shareholders are scheduled to vote on the Disney deal on July 10, but that date will be moved back if Mr. Murdoch and the Fox board decide to support Comcast’s offer. Disney would then have five days to respond with a counter bid.

Mr. Roberts wrote in a 700-word letter to Mr. Murdoch and his sons — James, the chief executive of Fox, and Lachlan, the executive chairman — that he “long admired what the Murdoch family has built,” but that he and his company were “disappointed” by Fox’s decision to engage Disney.