"He's in a coma, we're just waiting for the announcement of his death," a relative told the Aftonbladet newspaper.
"The doctors have said there is no hope."
The 26-year-old man was shot in the head while attempting to make a getaway from a robbery at a jewellery store in Södertälje, an hour south of Stockholm.
Police arrived before the thieves could escape, and a shoot-out in front of the store began, with the police hitting the 26-year-old in the head.
His accomplices left him on the ground and drove off amid the gunfire. The man's body and a weapon were left behind at the scene.
Six people between 19 and 26 years old were arrested over the weekend for the heist, and at least four have a criminal record. One of the suspects was released on Sunday afternoon, and another young woman was arrested for her suspected participation in the heist.
The 26-year-old thief, who is currently under continued surveillance by doctors at Karolinska Hospital, has been joined by family and friends from Norway and Germany, wrote Aftonbladet.
Relatives said the bullet is still lodged in his head.