Bauza told Cadena SER radio that two other victims were local residents, including an elderly woman whose body was found in the basement of her house.
The rain was so heavy a river burst its banks.
The Foreign Office has said: "We are in contact with the Spanish authorities responding to flooding in Majorca, and urgently seeking updates from them".
The town of Sant Llorenc is around 60 kilometres east of Palma - the capital of the Mediterranean island.
The latter were two British tourists who were in a taxi which was swept away by flood water.
"It all happened in less than 10 minutes", an unidentified witness with wet clothes and a blanket over his shoulders told TVE.
Most of the victims were discovered in the town of Sant Llorenc and another in S'illot.
Rescue workers and military units from the Spanish mainland were heading to the area which is popular with British holidaymakers.
At least 220 millimetres of rain are believed to have fallen in the region, flooding homes, roads and fields.
Dramatic video footage on social media shows raging torrents of water gushing through the streets of the town, with reports that more than 20 centimetres of rain fell in just four hours.
According to the Spanish weather agency, more heavy rainfall is expected in eastern and southern parts of the Iberian peninsula.
Spanish Prime Minister Pedro Sanchez tweeted his "solidarity and support for the relatives and friends of the victims, and for all those affected by this tragic flooding".