WASHINGTON—Special counsel Robert Mueller is investigating former White House national security adviser Mike Flynn’s work on an unfinished film financed by Turkish interests as part of its wider probe into whether Mr. Flynn improperly concealed financial ties to Turkey and to Russia, according to people familiar with the matter.

The focus on the film comes amid signs that Mr. Flynn may be working on a deal with Mr. Mueller. Mr. Flynn’s lawyers this week stopped cooperating with White House attorneys defending the president, according to other people familiar with the matter, a possible sign that Mr. Flynn is sharing information with investigators.

The Federal Bureau of Investigation is preparing to interview consultants hired by Mr. Flynn to work on a documentary film targeting an exiled Turkish cleric who Ankara accuses of trying to overthrow the country’s president, according to people familiar with the investigation. Details of the unfinished film project were first reported by The Wall Street Journal in May.

The film was at the heart of Mr. Flynn’s work for Turkish interests while he served as top adviser on then-candidate Donald Trump’s presidential campaign. While working to get Mr. Trump elected, Mr. Flynn’s company signed a $530,000 contract with a Turkish businessman with close ties to President Recep Tayyip Erdogan.

Mr. Flynn and his business partner, Bijan Kian, hired consultants to create a film attacking the Muslim cleric, Fethullah Gulen, who has been accused of orchestrating a failed July 2016 coup attempt in Turkey from his home in rural Pennsylvania. Mr. Gulen has denied playing any role in the short-lived putsch.