Detention warrant was issued on Friday for Fatih Samed Kılıç, son of Haşim Kılıç, a former chief justice, as part of an investigation into the Gülen group, the state-run Anadolu news agency reported.
Kılıç is accused of using a mobile application called Bylock.
Turkish authorities believe ByLock is a communication tool among followers of the Gülen group, which is accused by the Justice and Development Party (AKP) government of orchestrating the failed coup, a claim the group denies.
Tens of thousands of people, including civil servants, police officers, soldiers, businessmen and homemakers, have either been dismissed or arrested for using ByLock since the coup attempt.