Family members of Gordon Ross attempted to pay him a visit at his home in Stuart Park on Tuesday, August 21 but found that he was not at the address.
Since then, the 47-year-old has failed to return to home and has not made contact with friends or loved ones.
Gordon has previously visited various cities within the UK including Dundee and London, spending significant time in these areas without making his whereabouts known.
Local enquiries are now under way to establish if he has once again travelled outside Edinburgh.
However, officers are keen to speak with Gordon as soon as possible and anyone who can assist police with their enquiries is asked to come forward.
Gordon is described as white, 5ft 11ins tall with a skinny build, short dark brown hair, green eyes, glasses and a fair complexion.
A police spokesman said: “Given the length of time since Gordon made contact with friends or family we are eager to trace him as soon as possible and ensure he is safe and well.
“Anyone who remembers seeing Gordon in the past month and can help officers establish his whereabouts is asked to contact police immediately.”
Anyone with any information can contact Lothian and Borders Police on 0131 311 3131, or the charity Crimestoppers in confidence and complete anonymity on 0800 555 111.