It is noted that the case was opened against the employees of the magazine.
According to Hurriyet, the presidential administration's communications department has promised that the authorities will take diplomatic and legal steps.
Earlier, Charlie Hebdo published on his cover a caricature of Erdogan amid the crisis in relations between the two countries.
On October 16, an extremist beheaded a teacher near an educational institution in the suburbs of Paris.
The attacker was shot dead by the police.
The murder is associated with an incident at a school, where a teacher showed caricatures of the Prophet Muhammad published in Charlie Hebdo during a lesson.
Commenting on the tragedy, French leader Emmanuel Macron said that the country would not refuse to publish such cartoons.
Erdogan said about Macron's "anti-Islamic attacks", which are a threat to Turkey's national security.