The body of 36-year-old Andrew Bow was found at a house in the city last Wednesday.
The Police Investigations and Review Commissioner (Pirc) will examine how officers responded to initial concerns about the property's occupant.
In 2013, police in Edinburgh were criticised three times for the way they dealt with cases of vulnerable men.
The commissioner recommended additional training for officers.
A Pirc spokesman said: "The Police Investigations and Review Commissioner has been instructed by the Crown Office and Procurator Fiscal Service to carry out an independent investigation after the body of Andrew Bow, 36, was discovered at a residential address in Edinburgh on 23 March 2016.
"The investigation will focus on the initial police response leading up to the incident, which included a report of concern about the welfare of the occupant of the property.
"A report on the commissioner's findings will be submitted to the COPFS in due course."
A police watchdog will investigate how officers dealt with the case of a man found dead in Edinburgh eight days ago.