DALLAS—The state of Texas has launched an investigation into alleged fiscal improprieties at the state’s largest chain of charter schools.

Behind the probe: charges by the president of Turkey that the schools are part of a $500 million a year front to fund the revolutionary aspirations of a Turkish cleric he claims backed a recent failed coup.

The probe by the Texas Education Agency was prompted by a series of complaints filed by a Washington-based law firm hired late last year by the Turkish government to lead its case against Fethullah Gulen, a political enemy of Turkish President Recep Tayyip Erdogan.

Mr. Gulen, who lives in rural Pennsylvania, is on trial in absentia in Turkey on charges related to overthrowing the government. Thousands of Mr. Gulen’s followers in Turkey have been arrested or detained by Mr. Erdogan following the failed coup earlier this month.

Last week, a spokesman for Mr. Gulen denied any connection to the coup and said his goal is nonviolent reform in Turkey.