Manvir Singh, 21, from Bishopbriggs, East Dunbartonshire, was in Renfield Street, Glasgow city centre, at about 4.45pm on Wednesday March 19.
Officers are studying images that suggest he travelled to St Andrews, arriving at around 9pm that evening.
Police Scotland have said his disappearance is out of character and appealed for anyone with information about his whereabouts to contact them.
His father Harpal Singh yesterday urged anyone with information about his disappearance to come forward, saying the family were desperate to know he was safe and well.
He said: "Manvir is a much-loved son and is known as Sonu to family and friends. We haven't heard from him or seen him for more than two weeks and we are very worried about him.
"We know that Sonu got on a bus in Glasgow and travelled to St Andrews on Wednesday March 19 2014, but we don't know what happened to him after he got off the bus in St Andrews.
"We are keeping in touch with his friends in the hope that they may hear something from him - all we want to know is that he is all right.
"If anyone has seen Sonu or knows where he may be, please do get in touch with the police so that we know he is safe and well."
The student's mother Kirandeep Singh said: "Sonu, if you are watching this, we want you to know that you are not in any trouble and that everyone is worried about you."
Mr Singh was last seen by his friend in Renfield Street, near to Sauchiehall Street. He was wearing a navy blue hooded top with yellow markings on it, dark jeans and tan boots and was carrying a black bag.
The former Bishopbriggs Academy pupil, who has a 14-year-old sister Harneet, known as Honey, had left his belongings, including his phone, bank card and glasses, at home at around 8.30am on March 19.
He is in his third year at Glasgow University and had recently failed his exams, but had not told his parents. He told his mother his results had not been released.
She later said she was concerned he may not have wanted to disappoint the family.
However, friends in the dance group Punjabi No.1, of which he has been a member for more than 15 years, said there did not seem to be anything bothering him when they last saw him, a few days before he disappeared.
Police scoured CCTV footage in a bid to discover where he had gone. A possible sighting in Manchester was later confirmed by police not to have been of Mr Singh.
Superintendent Thom McLoughlin said: "Despite extensive inquiries we have still not traced Manvir.
"We are currently investigating the latest sighting of him in St Andrews and continue to actively seek new lines of inquiry. Manvir is a young, intelligent, active young man and there is nothing in his background to indicate why he would not maintain contact with his family.
"As time goes by we are all becoming increasingly concerned for his safety.
"The lack of contact has been extremely distressing for his family and I would urge anyone who recognises his description, or has any information regarding his whereabouts, to contact us."
Mr Singh is described as 5ft 10ins tall, of slim build with short black hair and a goatee beard.