Barrister Sayedul Haque Suman, a Supreme Court lawyer, filed the writ which is likely to come up for hearing at the bench of Justice Sheikh Hassan Arif and Justice Razik-Al-Jalil on Tuesday.
The writ also sought overhauling the security measures inside jail.
The Home Secretary, Inspector General of Prisons, Deputy Inspector General of Rangpur division and jailer of Panchagarh Jail were made respondents to the writ.
Palash Kumar Roy, 36, a Bangladesh Bar Council-listed lawyer and son of Pranab Kumar Roy of Borosingia village in Atoari upazila, sustained burn injuries in a toilet of the Panchagarh Jail on April 26 and succumbed to his injuries at the burn unit of Dhaka Medical College and Hospital on April 30.
Rajib Rana, a young man, filed a case against Palash with Sadar Police Station for reportedly making derogatory remarks about Prime Minister on March 25.
Later, police arrested him and a court sent him to jail after he produced before it on March 26.