WASHINGTON—Special Counsel Robert Mueller is investigating former White House national security adviser Mike Flynn’s alleged role in a plan to forcibly remove a Muslim cleric living in the U.S. and deliver him to Turkey in return for millions of dollars, according to people familiar with the investigation.

Under the alleged proposal, Mr. Flynn and his son, Michael Flynn Jr., were to be paid as much as $15 million for delivering Fethullah Gulen to the Turkish government, according to people with knowledge of discussions Mr....