The Junker JU52 HB-HOT aircraft, built in Germany in 1939 and now a collectors item, belongs to JU-Air, a company with links to the Swiss air force, the ATS news agency reported.
Police have called a news conference for 1200 GMT.
The Junker plane, which can carry up to 17 passengers and three crew, crashed into the Piz Segnas mountain in the east of the country on Saturday, at an altitude of around 2,500 metres (8,200 feet).
The flight had taken off from Ticino in the south of the country, Blick added, and had been due to land at the Duebendorf military airfield near Zurich on Saturday afternoon.
The 20 Minutes newspaper quoted a witness who was on the mountainside at the time of the crash.
"The plane turned 180 degrees to the south and fell to the ground like a stone," the witness said, adding that the debris was scattered over "a very small area," indicating an explosion was unlikely the cause of the crash.
Police had not provided an official toll by late Sunday morning, but said that five helicopters were involved in a search and rescue mission and the airspace over the crash site was to remain closed until late Sunday.
JU-Air said on its website that it was "deeply saddened" and its "thoughts were with the passengers, the crew and families and friends of the victims".
The company's flight operations were suspended, it said.
JU-Air says it runs a small fleet of four Junker planes, all built in 1939, which are for hire. Its pilots are ex-military and professional pilots, all of them volunteers.
On its website, JU-Air mentions one past accident, in 1987, at the Koblenz airport in Germany in which nobody was hurt.
In another Swiss plane crash on Saturday, a tourist plane carrying a couple and two young children crashed in a forest in the Nidwald canton and immediately burst into flames. No survivors have been found.
The vintage WWII plane crashed into a mountainside in the eastern canton of Grisons.