Bilel Ayadi, 34, who is in a medically-induced coma, has been unable to tell detectives how he sustained burns to his torso and neck.
He was found walking in Finsbury Park, north London, on 23 March.
Mr Ayadi had walked bare-chested for about two miles, police said.
He was found in Woodberry Grove at about 22:30 GMT by two security guards, who called police.
Appealing for information, Det Con Anwen Clissold said Mr Ayadi would have "been very distinctive".
"It may well be that he was already injured as he made his way through the streets," she said.
"We are still trying to establish what has caused his burns but what we do know is that a young man has sustained some very serious injuries. I need to find out how."
Mr Ayadi's brother, Abderrezak Ayadi, said: "Bilel is my brother and at this moment he is lying in hospital with terrible injuries.
"Imagine if this was someone from your family. We just need to find out exactly what happened to him."
Algerian national Mr Ayadi, who has lived in the UK for the past six years, is described as 5ft 11in (1.8m) tall and of medium build.
Images of a shirtless man found wandering the streets after he suffered serious burns have been released in a bid to solve the mystery of his injuries.