Officers were called to an address in Limefield, Edinburgh on Saturday morning.
The child was taken to the Royal Hospital for Sick Children where his condition is described as stable.
The incident has sparked a co-ordinated investigation, involving Police Scotland, NHS Lothian and Edinburgh City Council.
A Police Scotland spokesman said: "In line with multi-agency child protection procedures we are working closely with NHS Lothian and the local authority to fully explore the circumstances of how this child was injured. Investigations of this nature are sensitive and complex and it is being approached in an open minded manner to help us fully understand what happened, which may take some time.
"Police in Edinburgh are investigating after being alerted to the serious injury of a 14 month old child within an address in Limefield, Edinburgh on the morning of Saturday 8th February.
"The child was taken to the Royal Hospital for Sick Kids where he remains in a stable condition. Inquiries are at an early stage."