Weinstein Co. is exploring a sale or shutdown and is unlikely to continue as an independent entity, a person close to the company said.

The film and television studio’s board of directors and other people close to the company have been approached by possible buyers, people with knowledge of the talks said.

The talks come as Weinstein Co. mulls how to move forward after firing co-chairman Harvey Weinstein Sunday amid more than a dozen accusations against him of sexual assault and harassment.

The board had previously been considering appointing co-chairman Bob Weinstein, Harvey Weinstein’s brother, and President David Glasser to lead Weinstein Co. under a new name. That plan is no longer on the table, according to the person with knowledge of the potential sale.

In a statement, Bob Weinstein, whose title remains co-chairman despite his brother’s exit, said, “Our banks, partners and shareholders are fully supportive of our company and it is untrue that the company or board is exploring a sale or shutdown of the company.”

Four members of the Weinstein Co. board have quit in the past week. Bob Weinstein is one of three directors who remain, a person close to the company said.

In his statement, Bob Weinstein added, “Business is continuing as usual as the company moves ahead.”

It remains to be seen if a deal can be reached for the studio to be sold as a whole and continue operating under a new owner or owners.

If Weinstein Co. were bought whole and continued to operate, suitors might include hedge funds looking to have a U.S. entertainment asset, a person close to the company said.

Another possibility is that Weinstein Co. would be shut down and its library of movies and TV shows and other assets sold in pieces. Still another scenario is a breakup, where the company could be sold as three separate divisions: a movie studio, a television arm and a library of older titles that could be licensed to cable networks and streaming services, a banker who has worked with Weinstein Co. said. Of those, the TV division and the library are most valuable, that banker and others said.

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The company’s film-production unit, though the best-known part of the company to most consumers, has generated uneven returns and was most associated with Harvey Weinstein.

Last year, when Weinstein Co. explored unloading its television arm, which makes “Project Runway,” the division was cash-flow positive and seeking a sale price in the hundreds of millions of dollars, according to a person who advised the company on the deal. Potential buyers of the individual units could include a broad list including major media conglomerates.

A sale or shutdown would mark an ignominious end for Weinstein Co. and a shake-up of the independent film scene, where the studio has long been a powerhouse with Academy Award-winning pictures such as “The King’s Speech” and “The Artist.”

People close to Weinstein Co., however, said problems have been mounting since a New York Times article last week reported Harvey Weinstein reached financial settlements with at least eight women. Since then, others have stepped forward to accuse the mogul of misconduct or assault, including Gwyneth Paltrow and Angelina Jolie.

Mr. Weinstein in a statement to the Times apologized for how he had “behaved with colleagues in the past.” A spokeswoman for Harvey Weinstein said in a statement, “Any allegations of non-consensual sex are unequivocally denied by Mr. Weinstein.”

Several executives at the studio are weighing whether they want to remain amid the continuing drama, a person close to the company said.

CBS Corp. ’s pay-TV channel Showtime wants out of a drama that Weinstein Co. is developing called “Guantanamo.” “We do not intend to move forward with the current configuration of the project and are exploring our options,” a spokeswoman for the network said.

Apple Inc. this week ended plans for a series about Elvis Presley it was developing with Weinstein Co. as producer. Amazon.com Inc. said it is “reviewing our options” for two shows it was set to co-produce with Weinstein Co.

One of the Amazon projects has already been cancelled. A show set to star Robert De Niro and Julianne Moore and to be written and directed by David O. Russell will not go forward. A spokesperson for the three said, “we have decided together that it is best not to move forward with this show.” The loss of the show is the second major blow to Amazon’s video efforts this week. On Thursday it suspended the head of its studio Roy Price after details of a 2015 sexual harassment complaint against him were revealed.

Executives at Weinstein Co. have discussed delaying the release of the one remaining prestige film on its slate for 2017, “The Current War,” which would be released in November.

In his statement, Bob Weinstein said plans remain on track for the release of several movies from Dimension Films, a division he has long overseen. He didn’t mention “The Current War,” and his spokesman didn’t respond to a question about it.

Creators of the musical “In the Heights,” Lin-Manuel Miranda and Quiara Alegria Hudes, said this week they were asking Weinstein Co. to release film rights to the show, which the studio bought last year.

“ ‘In the Heights’ deserves a fresh start in a studio where I’ll feel safe,” Ms. Hudes said in a statement.

Hollywood agents say they are unlikely to suggest clients work with Weinstein Co. even without its former co-chairman, in part because the tumult makes it hard to predict if and when projects will be completed.

Goldman Sachs Group Inc. is exploring options for the stake it holds in Weinstein Co., which is now valued at less than $1 million, according to a Goldman Sachs spokesman. “There is no place for the inexcusable behavior that had been reported, and we strongly condemn it,” he said. The Weinsteins own 42% of Weinstein Co., with the remainder held by a variety of funds, banks and companies.

Fidelity Investments holds company shares in three of its funds, including its $118 billion Fidelity Contrafund, according to filings. Those shares account for 0.001% of the Contrafund.

Even before the revelations about Harvey Weinstein’s conduct, the Boston-based mutual fund giant’s view of the company’s health had deteriorated. Within the Contrafund, according to the filings, the value of Fidelity’s shares had by the end of August been marked down from their purchase price by 96%.

—Keach Hagey contributed to this article.

Write to Ben Fritz at ben.fritz@wsj.com and Erich Schwartzel at erich.schwartzel@wsj.com

Corrections & Amplifications

CBS Corp.’s pay-TV channel Showtime plans to move ahead with the drama “Guantanamo,” as long as Weinstein Co., which had been developing the series, is no longer involved. An earlier version of this article incorrectly suggested that the network was pulling out of the show. (Oct. 16, 2017)