Three-month-old Mohammad Sudais suffered third degree burns in a gas explosion at his home in Peshawar seven weeks ago.
The accident claimed the lives of his father Mohammad Ameen, 30, mother Sumaira, 28, and 13-month-old brother Abdul Haseeb.
Doctors in Pakistan say they can do no more for the baby, and have recommended that he be treated at Glasgow Royal Infirmary, in the city where his uncle Mohammad Asif lives.
Positive Action in Housing said the boy could be in Glasgow within 24 hours after his passport and guardianship papers are readied today.
The charity has appealed to the Scottish Government for help in arranging suitable transportation.
Director Robina Qureshi said: "Unless the doctors tell us different, we don't think a commercial flight is safe, possible or practical.
"If the Scottish Government brings the child from Pakistan by air ambulance or by a medical or even military flight with medical facilities, then there is nothing to prevent the child being in Glasgow's Yorkhill Hospital by this time tomorrow.
"The baby's uncle, Mohammad Asif, a leading member of the Afghan community in Glasgow, is desperate to do right by the child. For five weeks, he unsuccessfully tried to get the child medical treatment in Scotland. That's five weeks too long.
"That child can be here tomorrow if we all work together to make it happen."
A Scottish Government spokeswoman said: "The Scottish Government is continuing to do all it can to help bring baby Mohammad to Scotland for treatment."