Police said in a statement they had set up a special group to investigate the fire.
“Investigators examined the scene this morning," police said. "Investigations so far show a technical defect can not be entirely ruled out, at the same time, suspected arson is also being investigated as a cause."
The fire at the asylum seeker home broke out shortly after 8pm in the Eimsbüttel area of the city.
The mother, 33, along with her two sons, aged six and seven, died and five people were seriously injured.
Another 10 of the 30 people living in the apartment block were also taken to hospital.
A spokesman for the city's fire service said it was one of the worst fires for years in Hamburg.
Most of the inhabitants could be rescued relatively easily but when firefighters came to the top floor they found three bodies.
The mother and her children had tried to flee down the stairs, the spokesman said.
Investigations initially showed a defect with the electricity box on the ground floor may have caused the fire.
The fire service said the blaze itself was not large, but the smoke it caused was deadly. "Smoke drifted upstairs like in a chimney," the spokesman said.
Witnesses are asked to contact police on 4286 56789.