Hours of heavy rains overnight caused rivers to burst their banks, flooding homes with mud in Putumayo province.
An unknown number of people are missing. One army officer said the main local hospital was struggling to cope.
Mr Santos, who has travelled to the area, said troops had been deployed as part of a national emergency response.
The region's governor, Sorrel Aroca, told Colombian media that whole neighbourhoods had been buried.
Jose Antonio Castro, mayor of the provincial capital Mocoa, told Caracol radio that the town was "totally isolated", without electricity and water.
On arriving at the scene, President Santos told reporters, "My heart and the hearts of all

Colombians are with the victims of this tragedy."
The mudslides were caused by the rise of the Mocoa River and three tributaries, a representative of the National Disaster Risk Management Unit told AFP news agency.
Rescue services said 17 neighbourhoods had been affected.
Mayor Castro said his own house had also been destroyed.
"The mud is up to the roof," he told the radio station.
March was the country's rainiest month since 2011, according to the state meteorological agency.
Putumayo borders Ecuador and Peru.
Across Peru more than 90 people have died since the start of the year because of unusually heavy rainfall, which also caused landslides and flash floods.
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