There’s little question that Mr. Bewkes has the confidence of his board and the good will of investors: When he first took over the company in 2008, its shares traded as low as $14. But he lifted the stock price not by acquiring properties but by shedding them, making his company a natural takeover target. He could face skepticism from shareholders who want to know why the company’s ambitious growth plans are not already reflected in its stock price, which has risen considerably since Fox’s bid.

Time Warner still has two of the most coveted assets in media: HBO, which consistently churns out hit shows, and a group of valuable sports broadcast deals.

The big test for Time Warner will come this fall, when it unveils its growth strategy to investors. People briefed on the company’s plans say it intends to squeeze far more money out of HBO, both in the United States and abroad. In addition to a national advertising campaign, it is planning to sell streaming subscriptions, currently available only in Scandinavia, in a number of countries, and to expand its online presence in the United States too. Earlier this year, it licensed some HBO programming to Amazon in a multiyear deal believed to be worth about $300 million, but it held back such popular shows as “Game of Thrones” and “Entourage,” which could be sold in lucrative deals still to come.

Time Warner also intends to leverage its sports rights deals with the N.B.A., the N.C.A.A. and Major League Baseball to negotiate higher fees from cable providers to carry its TNT and TBS cable networks, according to people inside the company briefed on the plans who declined to be identified because the company has not yet disclosed its strategy.

From Time Warner’s point of view, the problem with Fox’s offer was not just its valuation of the company. It was the fact that 60 percent of the bid came in the form of Fox stock. Last month, Fox secured more than $9 billion in a sale of its Italian and German pay-TV business, and it could raise even more cash by borrowing money and by selling off most of the Turner networks, including CNN, HLN and TruTV.

It would still be a stock-heavy deal, though. As such, Time Warner is trying to highlight to investors the uncertainty surrounding the company’s leadership. Chase Carey, 21st Century Fox’s widely respected president and chief operating officer, is signed only through the end of 2015. Even if Fox extended his contract as part of the deal, the $150 billion company would probably soon be run by Mr. Murdoch’s 41-year-old son, James.

Image James Murdoch, co-chief operating officer of 21st Century Fox. Credit... Adrees Latif/Reuters

Mr. Murdoch has expressed confidence in his son’s ability to lead Fox. When he elevated James to co-chief operating officer of the company in the spring, he credited him with playing a major role in the growth of Fox’s global TV business.