But they have ruled out any form of foul play at this stage.
Police Scotland say the discovery is still being treated as “unexplained”.
But they confirmed there appeared to be nothing suspicious involved.
A major operation was launched last week, closing off streets and paths around the popular Glenogle Road area.
Forensic teams and specialist officers donned protective and waterproof gear after the alarm was raised around 7.20am a week past Thursday.
The man’s body was recovered from the river and stretchered to a waiting private ambulance.
His identity has so far not been released.
Today, Police Scotland said a report would be going to the Procurator Fiscal’s office in relation to the incident.
In a statement, the force said they were still looking into the circumstances.
It read: “Police in Edinburgh are investigating after the body of a 56-year-old man was sadly recovered from the Water of Leith in the Glenogle Road area on Tuesday 10 April.
“The death is being treated as unexplained but not suspicious and a report will be submitted to the Procurator Fiscal.”