The offer, first reported by The New York Times, was worth about $80 billion, or $85 a share, when it was made in June.
Fox estimates that a combined company, which would have $60 billion in annual revenue, would save $1 billion in costs and possibly more, the people familiar with the matter said. Detailed negotiations with Time Warner could reveal even more potential savings, they said, which may then justify sweetening the offer.
Fox said no talks were under way, and it has no desire to go hostile or to bid against itself with a higher offer, the people familiar with the matter said.
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Either way, a takeover by Fox could win the blessing of many of Time Warner's shareholders, a majority of whom also own Fox's non-voting stock. In addition, Time Warner does not have a staggered board and its bylaws allow as few as 15 percent of shareholders to call a special meeting, factors that could help a takeover bid.
A number of Time Warner shareholders said Twenty-First Century Fox and Murdoch might have to raise the bid to as high as $95 per share from $85 when the bid was made in June to wrap up a merger. Murdoch also would do well to offer more cash upfront, investors said.
Time Warner has no controlling shareholder, meaning the company could easily go into play, said Ken Griffin, founder and chief executive of Citadel, which owns shares in both companies. "It's going to be tough to say no," he said during a conference in New York.
Mario Gabelli, chairman and CEO of Gamco Investors, said he expects a technology company with more cash than Fox, such as Apple or Amazon, could emerge with a bid.
"It's not a dynamic I want to dismiss out of hand. They have a currency, and they understand the value of content," he said of the tech companies." Gamco's funds own about 3.6 million shares of Time Warner and about 10.5 million Fox shares.
Seeking dance partners
Fox's overtures are likely to reverberate across the industry, accelerating a wave of consolidation that is already well underway.
Comcast, the largest U.S. cable provider, offered in February to buy Time Warner Cable for $45.2 billion in stock. (Disclosure: Comcast is the owner of NBCUniversal, the parent company of CNBC and CNBC.com.) Overseas, Fox's 39 percent-owned British Sky Broadcasting Group is negotiating to buy Fox's Sky Italia and its Sky Deutschland subsidiary in a deal that could net Fox as much as $13 billion.