The only missing person as of Wednesday afternoon was a 5-year-old boy who disappeared with his mother.
Spain's meteorological office warned that rainfall and storms were set to get worse in the archipelago "in the coming hours".
The Britons, who were married and in their seventies, were travelling by taxi about 40 miles east of the Spanish island's capital, Palma, when the auto was swept away.
"The most important thing right now is to find the missing people and respond to the concerns of their families and to everyone who lives near the affected areas", he told reporters.
Videos shot on mobile phones by local residents showed a strong current of water and mud that buried cars and tore trees on its way down the streets of the town of 8,000.
At least three towns closed schools and hundreds of people were sheltered in sport facilities and a local horse racing track.
"In just two hours, some 180 litres of rain fell and we realized that we could not control the water", she said.
The thunderstorms are still circulating in the region and further torrential rain is expected on Wednesday and Thursday, with southern France and northwest Italy being most at risk of flooding.
Prime Minister Pedro Sanchez visited the rescue coordination centre and offered condolences for the victims.
Sanchez will be flying to the area later Wednesday, his office said.
Spanish lawmakers observed a minute of silence in parliament in memory of the victims. Streets in towns near Malaga on the Costa del Sol experienced chest-high water levels.