Mohammad Sudais suffered third degree burns in a gas explosion at his home in Peshawar seven weeks ago.
The accident killed 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 he be treated in Glasgow. The boy's uncle, Mohammad Asif, lives in the city.
Positive Action In Housing said the child could be in Glasgow later today after his passport and guardianship papers were readied yesterday.
The charity has appealed to the Scottish Government for help in arranging suitable transportation.
Director Robina Qureshi said yesterday: "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 is five weeks too long.
"That child can be here tomorrow if we all work together to make it happen."
A Scottish Government spokeswoman said: "We are continuing to do all it can to help bring baby Mohammad to Scotland for treatment."