The case was registered at the Andheri police station a few days ago based on a complaint filed by Nitesh Tiwary (27), general secretary of Andheri unit of the Bharatiya Janata Yuva Morcha (BYJM), police said.
The police, quoting the complaint, said one Anil Bhawiskar has uploaded an objectionable post about the prime minister's nonagenarian mother Hiraba Modi on a Facebook group page run by him.
The accused has been booked under relevant sections of the IPC and the Information Technology Act, the police said.
The Facebook profile seems to be fake and is suspected to be managed from somewhere in Uttar Pradesh, the senior official said.
"The profile has been blocked and we are trying to trace the person," he said, adding further probe in the matter was underway.