The vehicle overturned between the villages of Susch and Zernez in the Alpine canton of Graubuenden on Thursday afternoon and rescue teams responded to the scene, authorities said.
According to Swiss media, 48 people were on the bus, Sky News reported.
A spokeswoman for ANVR, a Dutch travel industry association, told the Blick newspaper that the passengers were all from the Netherlands.
Pictures from the scene published by the Swiss newspaper showed a bus on its side surrounded by helicopters and emergency response vehicles.
A police spokesperson said no other vehicles were believed to be involved.
Several ambulances and five helicopters were deployed to the location, according to Swiss media.
The Dutch Foreign Ministry said it was too early to confirm the number of victims.