Among those seeking euthanasia are elderly parents, siblings and children of the convicts.
"We request you (President) and the parents of the victim to accept our request and give us permission for euthanasia and stop any crime from happening in the future so that another incident like Nirbhaya doesn't happen and the court doesn't have to hang five people in place of one," the families of the convicts said in the letter in Hindi.
The families said that there are no sins that cannot be forgiven.
"In our country, even "Mahapapi" (great sinners) are forgiven. Revenge is not the definition of power. There is power in forgiving," the letter read.
This comes as the four convicts in the 2012 case -- Vinay Sharma, Akshay Singh Thakur, Pawan Gupta and Mukesh Singh -- are scheduled to be hanged at 5:30 am on March 20.
So far, President Ram Nath Kovind has rejected the mercy petitions of all the convicts in the matter. However, a "complete" mercy petition of convict Akshay Singh Thakur has also been filed claiming that the previous one, which was rejected by the President, did not have "complete facts".
Meanwhile, several petitions filed by the convicts are pending before Delhi court, Delhi High Court and Supreme Court.
The case pertains to the brutal gang-rape and killing of a 23-year-old paramedical student in a moving bus on the night of December 16, 2012, by six people including a juvenile in Delhi. One of the adults accused in the matter had allegedly committed suicide in the jail. The woman had died at a Singapore hospital a few days later.