The ruling in Switzerland comes after two Syrian brothers, aged 14 and 16, declined the traditional greeting with the female teacher, saying Islam forbids them to do so because she is not their wife or a member of their family.
The boys’ family, which has applied for Swiss nationality, denied accusations of radicalisation after national press reported an Islamic State (ISIS) propaganda video was found on one of their computers.
The father, an imam at a controversial mosque in the European country, has also had to deny allegations of manipulating his two sons.
A controversial exemption from shaking their female teacher’s hand at the beginning and end of lessons had been granted for the two teenage brothers.
The decision, which was considered “discriminatory against woman”, sparked outrage in Switzerland and represented a move away from the country’s traditional values.
But authorities have now ruled Muslim parents could face a fine of 5,000 Swiss francs (£3,430) if their children refuse to shake a female teacher’s hand.
The incident at a high school in Therwil, Bâle-Campagne, north Switzerland, led to a debate in the country, where handshakes are the traditional wave of greeting people.
The regional authority said teachers “had the right” to demand handshakes as a sign of politeness and respect.
In a statement the school board said: “Public interest concerning male and female equality as well as the integration of foreigners outweighs pupils’ freedom of religion and belief.”