According to Zurich police, three people picked up severe injuries, while the other casualties are being treated for minor and moderate injuries.
The crash occurred in the early morning in treacherous snowy conditions, police added.
The bus, operated by a subsidiary of the German-owned firm Flixbus, was heading to the German city of Dusseldorf, from Genoa, Italy, the ATS news agency reported.
Following the crash, a spokesman for the Russian Embassy to Switzerland, Stanislav Smirnov, told Sputnik that a Russian citizen who has suffered injuries as a result of a bus crash is in a serious condition.
"One Russian citizen is currently in a serious condition. She is not transportable. The rest of the injured Russians will be able to leave the hospital as early as today. They have minor injuries, there are also fractures," the spokesman said.
A source in the Russian Federal Agency for Tourism (Rosturism), in turn, told Sputnik that the agency was clarifying the information regarding the condition of the injured Russian nationals.
"There is information indicating that the bus was not a tourist one but a regular one," the source said, adding that it meant that the Russians who were on the bus were not traveling as part of an organized tour.
According to the police, there were also passengers from Albania, Benin, Bosnia and Herzegovina, Colombia, Germany, Ghana, Italy, Jordan, Nigeria, Romania and Switzerland on the bus. The driver is among those in a serious condition.