Portugal has sent medical teams, rescue workers, divers and relief supplies to the island, about 1,000km southwest of Lisbon, the capital.
Al Jazeera's Emma Hayward, reporting from Funchal, said there is an enormous rescue operation under way.
"The big task ahead will be trying to find any of those trapped survivors ... It will be very difficult to do so, the terrain of this island is pretty tricky," she said.
The Portuguese naval frigate Corte-Real set off from Lisbon for Madeira late on Saturday with helicopters to help with the emergency operation and other supplies, a military statement said.
"We are a mountain island, and water just came down the river carrying everything in its way"
Ana Basilio, reporter for the Lusa news agency
Two helicopters and two C-130 Hercules transport aircraft were also en route along with 89 police and firefighters.
Witnesses described how the mudslides filled some homes up to the second floor, while flash floods swept away bridges and buildings.
Television footage showed powerful streams of water and mud flooding the streets of Funchal, dragging and overturning cars and destroying trees.
Ana Basilio, a reporter for the Lusa news agency, said: "When it all happened, we couldn't see anything, we couldn't walk.
"People in the market had to be evacuated while shopping. People with cars couldn't move in the roads. Communications were all off, so people couldn't reach their families.
"We are a mountain island, and water just came down the river carrying everything in its way. There were cars, fridges and cupboards and even people. And we couldn't do anything."
At least 250 people have been made homeless by the heavy rains, the worst to hit the island since October 1983, when eight people died.
Jose Socrates, the Portuguese prime minister, has expressed "profound shock" at the disaster and promised support for those affected.
"I am absolutely saddened and shocked with the images, with the consequences of this calamity," he said, as he prepared to travel to Madeira to assess the damage.
Portuguese authorities have promised to organise aid for the autonomous region, and the country's government has said it is considering asking the European Union for help with the crisis.
Officials said they are also discussing whether to declare a state of emergency on the island.
Madeira is the main island of an archipelago of the same name off the northwest coast of Africa.