Discovered in the grounds of Ballumbie Primary School shortly before midnight on Saturday, Police Scotland suspect the body is that of Allan Smeaton, 49, who had been missing since the end of March.
Officers have informed his family of the discovery.
Mr Smeaton was last seen in the early hours of Saturday, March 28, with his concerned family urging him to return home and issuing a public statement to tell him: "Everyone is missing you."
A Police Scotland spokesman said: "Police Scotland can confirm that the body of a man was found within the grounds of Ballumbie Primary School, Lothian Crescent, Dundee, at around 11.45pm on Saturday, April 11.
"The family of local man Allan Smeaton, missing since the end of March, have been made aware of the discovery.
"Formal identification is yet to take place and a post-mortem will be carried out in due course to establish the exact cause of death. The death is not being treated as suspicious."