The 10-month-old boy was being looked after at Bright Horizons Nursery in Corstorphine when the incident happened on Tuesday.
He was taken to hospital but died the next day.
A spokeswoman for Police Scotland said: "Police in Edinburgh were contacted after a 10-month-old boy took unwell within a private nursery in the Corstorphine area of the city on Tuesday.
"The child was taken to the Royal Hospital for Sick Children where he sadly passed away on Wednesday.
"The death is being treated as unexplained, but not suspicious, and a report will be submitted to the procurator fiscal."
The nursery is closed and the families affected are being offered care at other providers.
A spokesman for the nursery said: "We are devastated by this news. Our thoughts are with the family at this time.
"Obviously we are co-operating fully with the authorities. The safety and well-being of the children in our care is our absolute priority.
"The nursery will be closed while we support our staff and families. In the meantime we have arranged for our families to be offered alternative care at neighbouring nurseries."