Spain's National Police confirmed on social media that five people had been killed in the heavy rains in the municipality of Sant Llorenc, and urged people to avoid driving in the Llevant area on the east of the island.
A former mayor of the town of Arta near Sant Llorenc des Cardassar which was worst affected by the floods has also been named as one of the five fatalities.
At least another 20 people were still missing after the downpours, local media reported, though emergency services would not confirm that figure.
A destroyed auto and debris are seen on the street as heavy rain and flash floods hit Sant Llorenc de Cardassar on the island of Mallorca, Spain, October 10, 2018.
Spain's weather agency said up to 22 centimeters (8.7 inches) of rain fell within four hours, triggering flash floods in the island.
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At least three towns closed schools and hundreds of people were given shelter in sport facilities.
The two British citizens were travelling in a taxi when a river burst its banks on the popular holiday island.
Public broadcaster TVE showed footage of flooded streets and overturned cars along the guardrails of a major road.
Authorities said 80 soldiers and seven vehicles from the military's emergency unit had joined Wednesday more than 100 rescuers who had been working in the area overnight.
Sanchez was planning to travel to the area later today, his office said.
Spanish lawmakers held a minute's silence before opening Sanchez's weekly question time.