The couple, who were in their 70s, are said to have died along with cab driver Juan Sillero when the car was swept away by a torrent of water near the coastal resort of S’Illot.
Authorities on the popular tourist island said 10 people, three of them foreigners, died after torrential rain deluged the region around the town of Sant Llorenc, about 40 miles east of the capital, Palma.
Emergency services assisted by the military were said to be also searching for a five-year-old boy missing since the disaster.
Some parts of the island received up to 9in (23cm) of rainfall in a matter of hours on Tuesday evening.
Witnesses described having minutes to find safety as the water levels rose, inundating houses and catching motorists off guard.
The floodwaters tore down trees, swept numerous cars and trucks into piles of twisted metal and buried streets under mud.
Mr Green, 77, and his wife, 75, were being driven to a hotel in Cala Bona when the taxi was caught by the floodwaters, Diario de Mallorca reported.
Rescue workers reportedly found the car submerged and on its side at the mouth of the torrent in S’Illot at 2am (1am BST) on Wednesday.
Military divers found the couple’s bodies still inside, while Mr Sillero was later found lifeless nearby, the paper said.
A Foreign Office spokesman said: “We are supporting the family of two British people following their deaths in Spain, and will do all we can to assist them at this deeply difficult time.
“Our staff remain in contact with the Spanish authorities who are responsible for responding to the floods, and are ready to assist any other British people who require our help.”
According to Diario de Mallorca, the bodies of an 81-year-old man and 83-year-old woman were among the first to be found on Tuesday.
Rafel Gili Sastre, 71, a former municipal mayor, was later found dead at his home, the paper said.
Authorities said 80 soldiers and seven vehicles from the military’s emergency unit had joined more than 100 rescuers deployed in the area on Wednesday.
Meanwhile tennis star Rafael Nadal, who is from the island, was seen joining in with the clean-up effort.
The world number one invited locals to take shelter at his tennis academy in nearby Manacor and then donned wellington boots to join other volunteers.
“Sad day in Majorca,” the 30-year-old said on Twitter.
Prime Minister Pedro Sanchez called news of the flooding “devastating” as he visited the area on Wednesday.
He said: “My solidarity and support goes out to the families and friends of victims and all the affected by these tragic floods,” he said.