“As Bob has said, ESPN is an integral part of the Walt Disney Company, and he believes that its full potential will continue to be realized,” said Disney spokeswoman Kristina Schake on Sunday.

Last month, Mr. Loeb’s hedge fund, Third Point LLC, said it had renewed its stake in Disney stock after having liquidated one earlier this year. He sent a letter to Mr. Chapek asking for major changes to Disney’s business, including spinning off ESPN, refreshing Disney’s board and cutting spending.

“We have a better understanding of ESPN’s potential as a stand-alone business and another vertical for [Disney] to reach a global audience to generate ad and subscriber revenues,” Mr. Loeb wrote on Twitter on Sunday. “We look forward to seeing [ESPN chief James] Pitaro execute on the growth and innovation plans, generating considerable synergies as part of The Walt Disney company.”

Mr. Loeb declined to comment beyond his tweets, a spokeswoman said, adding that the rest of the requests that Mr. Loeb made in his letter to Disney still stand. Messrs. Loeb and Chapek have regular conversations and are currently in close contact, people familiar with the matter said.

The new stake for Mr. Loeb’s fund represented less than 1% of Disney’s shares outstanding and at the time had an economic value of around $1 billion, The Wall Street Journal has previously reported.

Disney shares are down 26.5% this year as of the close of trading Friday. In August, the company reported stronger-than-expected financial results and an addition of 14.4 million subscribers to its Disney+ streaming service.

Last month, Disney said in response to Mr. Loeb’s Third Point letter that it welcomes the views of all of its investors. The company said its board has been continuously refreshed, with an average tenure of four years.

The idea of selling ESPN—which sends a steady flow of cash to Disney via licensing agreements with cable-TV operators—has come up frequently in recent years as the price of sports broadcast rights has steadily risen. Some investors have argued that ESPN is more valuable as a stand-alone company than as a division of Disney.

While fewer Americans are getting cable TV, the sports network is home to some of the most-watched events on television and it attracts huge numbers of prime-time viewers, according to Nielsen. ESPN also brings in significant fees to Disney. On average, every American with a pay-TV package that features ESPN pays more than $100 a year for access to the network, according to Kagan, a media research group within S&P Global Market Intelligence.

For Disney, ESPN+, the streaming service attached to the sports network, is growing. ESPN+ has 22.8 million subscribers, Disney reported last month, a 53% gain from a year earlier.

Mr. Loeb’s August letter pushed for Disney to do another deal: to make every attempt to buy up

Comcast Corp.’s

remaining minority stake in the streaming service Hulu before a sale can be forced by Comcast as early as January 2024. Under a 2019 agreement, Comcast can require Disney to purchase its NBCUniversal subsidiary’s one-third stake in Hulu for at least $9 billion, assuming the streaming service has an equity value greater than $27.5 billion.

Third Point said that the Hulu deal is of urgent importance and that it has been made more feasible by the recent decline in the share prices of streaming video businesses such as Netflix, the Journal previously reported.

Some other investors say if Disney waits to buy Hulu, it risks paying a higher price than they say it might pay now.

“We believe that it would even be prudent for Disney to pay a modest premium to accelerate the integration but are cognizant that the seller may have an unreasonable price expectation at this time,” Mr. Loeb’s letter said.

Mr. Chapek said in an interview published this weekend in Variety that he has been in continuing discussions with Comcast to buy the rest of Hulu. A Comcast spokesman declined to comment.

Hulu has been a boon to Disney in part because the service provides more adult-oriented shows and films, drawing subscribers to both Hulu’s stand-alone streaming service and Disney’s streaming bundle, which includes Disney+, Hulu and ESPN+. Mr. Chapek has said that the company wants to invest in more general entertainment content, and has highlighted hit series such as Hulu’s “Pam & Tommy” and “The Dropout.”

Last month, Disney said the premium Disney streaming bundle, which includes ad-free versions of Disney+ and Hulu, as well as a version of sports-focused ESPN+ with ads, will remain at its current price of $19.99 a month in the U.S., while a bundle that includes all three services, but with ads on Hulu, will rise in price by $1 a month, to $14.99.

Mr. Loeb is one of Wall Street’s best-known agitators and is known for fiery letters and campaigns at companies that often involve calling on management to consider strategic changes such as separating units or exploring sales.

Third Point urged Intel Corp. to consider selling some acquisitions or splitting its design and manufacturing operations in late 2020. The chip maker replaced its CEO a few weeks later, a move that Mr. Loeb applauded. In addition to making a few activist bets most years, Third Point also makes other hedge-fund and nonactivist investments.

Write to Robbie Whelan at robbie.whelan@wsj.com