Police, Edinburgh City Council and NHS Lothian are working together on the case after officers were called to an address in the Limefield area on Saturday.
The boy was taken to the Royal Hospital for Sick Children where it is understood he is being treated for a brain injury.
A Police Scotland spokesman said last night that no one had been detained or arrested in connection with the incident.
He said officers were investigating "various" possibilities, including that the child had fallen or been shaken. A police car remained parked outside the address yesterday.
A spokesman said: "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.
"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.
Neighbour Carol Crawford, 69, who described the boy as a "quiet" child, said: "A paramedic came out of the house holding the baby.
"I have only seen the baby there once before. All sorts of things are going through our heads now. We just don't know what has happened and nobody has told us anything. Our heart goes out to the family."