A man matching the description of French 21-year-old Antoine Maury, who studies music and business at Edinburgh College, was spotted by a dog walker, who said he seemed “agitated”.
The sighting took place around 10.50pm before a search was launched after a mobile phone and clothes were discovered at Duddingston Golf Club the next day.
Mr Maury’s family has now flown in from France as officers continue their searches with police dogs and specialist rescue units.
A poster appeal has also been launched.
Mr Maury is described as white, around 6ft, of medium build, with short brown hair, a beard and a fair complexion. He has a tattoo of a bleeding hand on his forearm.
He was last seen wearing a brown/green bomber jacket with dark chinos and a white beanie hat. He is thought to have been wearing black and white Vans trainers.
Inspector Alan Carson said: “We’ve no concerns over Antoine’s frame of mind or him setting out to do anything untoward but it is a question we still have. We’re obviously concerned in that regard with the fact the man was running topless and him possibly suffering the effects of the temperature and not being able to deal with that.
“We have no indications what Antoine’s intentions were as he was walking in the opposite direction of his student accommodation. There was no indication of any fallout with his friends.
“There is nothing to suggest there was any acrimony between him and his friends.
“We’re unable to confirm whether or not Antoine had consumed alcohol but clearly as he was socialising with friends it’s something we’re keeping an open mind about.
“There is no suggestion of foul play, however equally we don’t know if he’s arranged to meet someone.
“If there is a possibility that he has friends elsewhere in the city he has arranged to spend a night or a few nights with we are keen to trace him and make him aware of the police campaign to find him.
“His family – mum, dad and sister – are now in the city and are assisting in any way they can in the investigation.”
Antoine holds dual Finnish and French nationality and spent time in both countries before moving to Scotland. He lives in Edinburgh College accommodation.