Ostergaard, 20, has not been seen since she left her apartment in the city of Zug on the morning of March 26th, when she was apparently planning to travel to the University of Zurich.
Cantonal police did not provide detailed reasons why they suspect she killed herself.
But Danish news site BT reported that the uncle of the woman was informed by police that a witness saw her walking along the shoreline of Lake Zug on the day of her disappearance.
Police have not confirmed this information.
But 15 police divers from all over Switzerland searched the lake on Thursday for the Dane, who was known in Zurich as a part-time model.
The search is set to continue on Friday with Zurich cantonal police specialists who plan to use an underwater camera to probe the lake bottom.
The hypothesis of suicide emerges after an extensive search of the Zug area and its two lakes with a helicopter, boats and dog tracking teams.
Police say they have checked Ostergaard’s mobile phone and bank records without gaining any clues as to her activity since she was reported missing last Wednesday.
The force came under criticism for not checking surveillance camera videotape from the Zug train station in a timely enough manner.
Swiss Federal Railways routinely erases videos 72 hours after they are taken and police did not put in a request to view footage until it was too late.
In a statement issued late Wednesday, cantonal police said an internal investigation was under way into why that happened.
But police insisted that not a single witness had reported seeing Ostergaard in a bus, in a train or at the Zug train station.
Searches by family and friends around Ostergaard’s home also failed to turn up any evidence.
Anyone with possible information is asked to telephone 041 728 41 41.