“At a time of dynamic change in the entertainment industry, the combination of Disney’s and Fox’s unparalleled collection of businesses and franchises will allow us to create more appealing high-quality content,” Mr. Iger said in a statement.

Disney’s increased offer comprises an equal mix of cash and stock: $35.7 billion in cash and 343 million shares in Disney. Fox shareholders can elect to receive either $38 in cash for every Fox share or Disney stock at an equivalent value. Disney plans to offer a ratio of its shares for every Fox share to make sure investors receive a value of $38 per share, what is known as a collar.

Fox has postponed its scheduled July 10 shareholder vote so that investors will have time to consider the new bid. A new date has not been set.

If a sweetened Comcast bid materializes, the Fox board will have to calculate not only the dollar value but the likelihood that the deal would pass government inspection. Both Disney and Comcast would face antitrust scrutiny, which is part of the merger calculus that Mr. Murdoch will have to make before choosing an eventual winner.

Comcast made its $65 billion offer a day after a federal judge approved a merger between AT&T and Time Warner. Comcast executives had seen AT&T’s court victory as a blueprint for the case it would make to regulators, since it would be challenged on two fronts: as a distributor of content and as a producer of content via its NBCUniversal division.

Mr. Iger disagreed with the notion that the AT&T ruling had paved the way for other big mergers. “It’s just simply an apples-to-oranges comparison,” he said on a call with analysts Wednesday, citing Judge Richard J. Leon’s warning that any attempt to apply his decision to future deals should be avoided.

The bad blood between Disney and Comcast goes back to at least 2004, when Comcast tried to swallow Disney whole. The Disney board fought off that attempt, but Mr. Iger and his top lieutenants have not forgotten it.