The death will be referred to the coroner.
A police spokesperson said emergency services were alerted to reports a baby was unresponsive at a Barrington Street property in Spreydon shortly after 2.30pm.
Medical attention was given but the 9-month-old died.
Oranga Tamariki referred all comment to police and refused to say whether they had any involvement with the family.
"As this is a police-led investigation ... all enquiries are best directed to them," Oranga Tamariki said.
In a statement, Kāinga Ora regional director for Canterbury Liz Krause said the agency's thoughts were with the friends and whānau of the baby who died.
"While this death is under police investigation, it would be inappropriate for us to comment further."