Mr. Trump also has connections to Mr. Singer, the founder of Elliott Management. He was once a staunch never-Trumper who raised money during the 2016 for Mr. Trump’s political rivals. Soon after the election, Mr. Singer got on board, contributing $1 million to the inauguration. In a February 2017 press briefing, President Trump said that he had just met with Mr. Singer at the White House, adding that the billionaire “has given us his total support.”

Elliott Management’s letter included data and insights from a wide variety of sources. As part of its due diligence, the hedge fund said, it has commissioned a consulting firm to survey “the wireless, pay TV and broadband preferences of more than 35,000 consumers to understand evolving consumer trends” and “more than 200 former AT&T and industry executives to accurately evaluate the current situation.”

Elliott Management went on to ask the company to stop striking new acquisitions, increase dividends and share buybacks and improve its efficiency. The fund also said it was seeking seats on AT&T’s board. Shares in AT&T rose about 1.5 percent on Monday.

While it has battled its main telecommunications rival, Verizon, in a bitter price war that has made each customer less profitable, AT&T, led by the chairman and chief executive Randall Stephenson, has acted decisively when it has come to WarnerMedia.

It put a veteran AT&T executive, John Stankey, in charge of the new division. Eight months into his tenure, the longtime HBO chief executive Richard Plepler, a gregarious figure who was known for having a way with Hollywood talent, stepped down. People familiar with Mr. Plepler’s thinking said at the time that he found that he had less autonomy as an AT&T employee. HBO had won 160 Emmys on Mr. Plepler’s watch.

Mr. Stankey named Robert Greenblatt, the former head of entertainment at NBC and Showtime, as the new chairman at WarnerMedia, a job that gave him oversight of properties like HBO, TBS and TNT.

Mr. Stankey has trimmed costs and is looking to increase profits through new business lines. In addition to unlocking higher returns on advertising, he has spent considerable time and money to create HBO Max. The service may help AT&T offer special discounts to its customers, potentially keeping them from defecting to other services.