Emergency services were called to an address in the Muirhouse area of the city this morning where the child was found seriously injured - but the toddler was pronounced dead a short time later.
Pictures from the scene in Muirhouse Place West show the entrance to a block of flats taped off and a police officer standing outside the door. Forensics officers are also in attendance.
Chief Inspector David Happs, from Drylaw Police Station, said: "Around 9.30am on Saturday, 21 November, 2020, officers and emergency services were called to an address in the Muirhouse area of Edinburgh where a two-year-old child was found seriously injured. He was pronounced dead at the scene a short time later.
"A 40-year-old man has been arrested in connection with this incident and enquiries are at an early stage and ongoing."
"We understand an incident such as this can cause distress and alarm to the local community. There will be a continued police presence in the area as we conduct enquiries."
"Anyone with information that may assist the investigation should report this to police on 101, quoting incident number 1010 of 21 November."