A fracture in the relationship between Sumner Redstone, the mercurial 84-year-old mogul who controls CBS and Viacom, and his daughter has become the latest drama within the House of Redstone, and how it plays out could decide the future control of one of the world’s great media fortunes.

Just months after settling litigation with his son Brent that resulted in Sumner buying out his son’s stake in the family business, he is now enmeshed in an ugly falling out with his other child, daughter Shari, and she is expected to leave Viacom’s board. The trigger for the latest Redstone family feud, according to several sources, was a dispute over corporate governance issues – such as Shari’s effort to more closely tie executive compensation to performance—at Viacom (VIAB) and CBS (CBS), a media empire that encompasses MTV, the CBS broadcast network, the Paramount film studio and the pay-TV channel Showtime.

“Shari is a committed and involved board member and she has no intention of leaving the board,” said Nancy Sterling, a spokesperson for Shari Redstone.

Fortune has learned that Redstone and his representatives (including Viacom CEO Philippe Dauman, Sumner’s longtime personal attorney) have been in discussions with Shari, his one-time heir apparent, and her lawyers about reducing her influence in the affairs of CBS and Viacom, where she sits on both boards as non-executive vice chairman. This could include a deal in which Sumner acquires Shari’s voting stock in Viacom and CBS, according to multiple sources close to the matter, although it is far from clear that Shari will assent to such an agreement. Under one possible scenario, according to a source familiar with the matter, Shari would get control of the family movie theater business in exchange for her stock in Viacom and CBS.

While her father oversees his empire from his Beverly Hills mansion, Shari continues to run the Massachusetts-based movie theater chain National Amusements—which has been in the family since 1936 and was the embryo for Sumner’s global media conquest. Gaining Shari’s voting stock — which amounts to close to 16 percent of the votes in both Viacom and CBS and is held by National Amusements — would simply add to Sumner’s already ironclad control of both companies.

Brent, in his lawsuit, had sought to dissolve the family media empire and cash out his one-sixth stake in National Amusements. Details of the settlement were not disclosed, but Brent’s share had been valued at about $1.3 billion. Sumner also faces a lawsuit from his nephew, Michael Redstone, who claims he was cheated out of a chunk of the family business.

After Brent was shuffled aside, Sumner was left holding 80 percent of National Amusements and Shari the other 20 percent. National Amusements in turn controls about 79 percent of the voting stock in Viacom and 76 percent in CBS—and about 11 percent of the equity in each. National Amusements has been valued at about $8 billion, meaning Shari’s stake is worth close to $1.6 billion.

Through spokespeople, Sumner and Shari declined comment.

The latest behind-the-scenes maneuvering over control of the Redstone empire is striking, given that in recent years Shari had taken on a more prominent role at the companies, and often attended management meetings at the company’s headquarters in New York. Sumner even so much as designated her as his successor, saying in a Wall Street Journal story two years ago that she would likely take over as controlling shareholder “after I’m gone.”

But more recently, the personal animosity between the two has ratcheted up and Sumner has sought to undermine her ability to be an effective board member. Notably, mentioned one source, when Shari was honored by the USO of Metropolitan New York in April at the group’s annual woman of the year luncheon, Dauman and Shari’s mother — Sumner’s first wife — Phyllis were in attendance, but not Sumner.

If Shari were to relinquish her share of Viacom and CBS, it would add more urgency to the question already bandied about in media circles: Who would become the controlling shareholder after Sumner? People in the media industry like to speculate that one possibility is Paula, his wife, who some say has taken on an increasing interest in the business—although that idea was dismissed by sources familiar with the matter. (Sumner has admitted that last year he took Paula’s guidance when he made the now-infamous decision to fire Tom Cruise at Paramount.)

[fortune-brightcove videoid=4735892119001]

Shari’s role within Sumner’s media businesses began to be more prominent in 2005, when she was appointed vice chairman of Viacom, after serving since 1999 as president of National Amusements. That year, frustrated by Viacom’s stagnant share price, Sumner decided to split the conglomerate into two publicly traded companies: CBS, which includes the broadcast network, a radio unit and Showtime; and Viacom, which houses MTV and the Paramount movie studio. During the transition, Shari played a major role in constituting the two boards of directors and ensuring that each had an adequate number of independent voices.

And herein lies the genesis of the fallout between Sumner and Shari. According to these sources familiar with Shari’s contributions to Viacom and CBS, Sumner became upset over the configuration of the boards, particularly at CBS. Eventually, two members that Shari had recruited to CBS, Ann N. Reese, a former CFO of ITT Corporation, and Judith A. Sprieser, former CEO of software company Transora, left last year.

For a time CBS, which after the board resignations had four independent directors out of eight total, was not in compliance with New York Stock Exchange regulations that require a majority of independent board members. (Companies with controlling shareholders are not required to adhere to such rules. But if a company voluntarily decides to be held to those standards, as Viacom and CBS have, they then are legally required to abide by the regulations.) This year, CBS added five new board members, which a source said were largely chosen by Sumner, including producer Leonard Goldberg, a frequent dinner companion of Sumner’s.

“He clearly had encouraged Shari to put together new boards,” said one of these sources. “How their falling out took place, when it took place, I just have no clue.

“It is hard to know what motivates that man. The most difficult thing for anyone to understand is his motivation.”

Related: Sumner Redstone’s Anti-Aging Secret (Fortune, 2007)

Another source said that “Shari was quite supportive of her father, but that wasn’t returned. Shari was an advocate of more modern governance rules.”

In particular Sumner balked at Shari’s push for the use of more performance-based measures to set executive compensation, these sources said. Yet in March, CBS said Sumner, who is executive chairman, agreed to a revised compensation package that was tied to shareholder returns. He has a similar structure in his compensation agreement with Viacom. (Last year CBS paid Sumner about $12 million, and Viacom paid him a package worth about $16 million.)

Ever since Sumner re-asserted control last summer with the Labor Day ouster of one-time favorite Tom Freston as CEO of Viacom, Shari has largely receded from any visible role, instead focusing on operating the 120 movie theaters owned by National Amusements.

The two have disagreed publicly over the future of movie exhibition, with Sumner saying that he sees little growth in the theater business, and Shari taking the opposite view. He has said he wishes Shari would sell it. And in a tiff that was only covered by the Boston papers, in April Shari objected to Sumner’s plan to cash out some stock held by National Amusements to make a charitable donation in his own name. Shari had felt that she should also be allowed a stock distribution to make a donation, or that Sumner’s gift should have been made in the name of National Amusements.

Sumner has been notoriously flighty on the topic of succession, and a cavalcade of favored executives have come and gone, always leaving with hefty severance payouts. In 1996 Sumner’s right hand man Frank Biondi was pushed out. In 2000, Dauman and Tom Dooley exited Viacom with reported payouts of about $150 million each. Three years later, when Mel Karmazin left, Freston was Redstone’s first pick for CEO, but he split the job after Freston demurred and CBS boss Les Moonves resisted the idea of reporting to Freston. Lastly, Dauman and Dooley were ushered back in after Freston was ousted.

Post-breakup, the performance of the two companies has not been what Sumner had envisioned. Viacom was designated the growth company, but its share price has lagged that of CBS. Since the two stocks’ debut in January 2006, CBS is up about 35 percent, while Viacom has gained just 1.5 percent.

Meanwhile, a parsing of Sumner’s public comments these days suggest a vast chasm between him and his daughter. In an interview earlier this year on CNBC’s “Conversations with Michael Eisner,” he said he won’t give her the CEO job. “Now, you wanna give away what you have to your family be my guest,” he said. In the same interview, he said, “my wife is closer to me these days than my daughter.”

More recently Sumner, also on CNBC, said Shari is a “viable candidate” to succeed him, but added it would be up to the board. But that is an evasive maneuver: Be wary when a controlling shareholder puts distance between himself and the board when it comes to making tough decisions. These days, and for the foreseeable future, there’s only one person making the big decisions at CBS and Viacom: Sumner Redstone.