The incident happened on Tuesday at the group’s Corstorphine nursery, which is believed to be based inside a David Lloyd centre.
The boy was taken to the Royal Hospital for Sick Children in Edinburgh for treatment, but died the following day.
The child was reportedly being fed mango when he got into difficulty.
On Friday it was reported that the 10-month-old had been named locally as Fox Goulding.
A spokesperson 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 wellbeing of the children in our care is our absolute priority. The nursery will be closed whilst we support our staff and families. In the meantime we have arranged for our families to be offered alternative care at neighbouring nurseries.’
A Police Scotland spokesman said, ‘The death is being treated as unexplained, but not suspicious, and a report will be submitted to the procurator fiscal.’