Antoine Maury was last seen in Edinburgh on Monday 24 October.
The 21-year-old was last seen at Edinburgh College's Milton Road East campus at 22:00 when he suddenly left his friends.
Between 22:45 and 23:00, a man matching Mr Maury's description was seen near Duddingston Village.
He was topless and heading towards Holyrood Park.
Police now also want to speak to people who were in the areas: Duddingston Village, Duddingston Loch or Prestonfield golf club the following day, Tuesday 25 October, who may have seen or heard anything unusual.
Insp Alan Carson, of Police Scotland, said: "We are interested in speaking to anyone who was in Holyrood Park between the hours of 22:30 and 23:30 on Monday 24 October, whether they saw Antoine or not.
"Equally, we are keen to hear from anyone that was in the areas around Duddingston Village, Duddingston Loch or Prestonfield Golf Club the following day, Tuesday 25 October 2016, who may have seen or heard anything unusual.
"Even the smallest amount of information may be the key to helping us find him.
"This is still a live investigation and we will carry on our search until we have exhausted our search parameters."
Mr Maury is white and of medium build. He is about 6ft, with short, brown hair, a beard and a fair complexion.
He has a tattoo of a bleeding hand on his forearm.
When last seen he was wearing a brown/green bomber jacket with dark-coloured chinos and a white beanie hat.
He is also believed to have been wearing black and white Vans trainers.
Glasses belonging to Mr Maury, who suffers from poor eyesight, have since been found with some of his clothes on a golf course.
Divers have been searching Duddingston Loch and there are police searches in Holyrood Park.
The police will search Edinburgh for a missing French student until they have "exhausted" all their "search parameters".