"My solidarity and support goes out to the families and friends of victims and all those affected by these tragic floods", said Sanchez.
"Our most honest condolences to the loved ones of the victims of the serious floods in San Llorenc", he wrote.
Earlier on Wednesday, Nadal said on Instagram he was offering to open up his sports centre and tennis academy to people made homeless by the floods.
Images posted on social media showed cars being swept along narrow streets by torrents of water.
Other victims were found in Arta and the coastal village of S'illot, where the creek that overflowed reaches the sea.
The regional government called an emergency meeting and authorities said rescue workers and military units were heading to the area to help.
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"It all happened in less than 10 minutes", an unidentified witness with wet clothes told the public broadcaster TVE.
Another resident, Tomas Oliver, cleaned his house as he celebrated that all the damage in his household had been material.
"We are grateful that we are all alive, but we pray for those that have lost their lives to this huge flood", Oliver said.
Rivers burst their banks, and swamped homes - forcing some people to take shelter in a sports center in the nearby town of Manacor. The flooding also affected phone communications, destroyed two bridges and forced the closure of 11 roads.
Muddy waters rushed down roads, houses were flooded and vehicles piled up on top of each other after the Mediterranean island was pounded with rain in just a few hours, video footage showed.
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.