The landslides hit a well-known tea plantation east of Colombo in the early morning and some homes were buried in 30 feet (nine metres) of mud, officials said.
Hundreds of soldiers were digging for survivors, but rescue efforts were hampered by damage to roads which prevented earth-moving and other heavy equipment from getting through.
"We have reports of 140 houses getting washed away in the mudslides," Sarath Kumara, a spokesman for the national Disaster Management Centre, told AFP.
"The latest we have got is that at least 300 people may be missing," the official said, updating an earlier estimate of 200.
Kumara said 16 bodies had been recovered by noon after the disaster in the Koslanda region, around 200 kilometres (125 miles) east of the capital.
The victims were tea plantation workers and their families, whose homes were located on a mountain slope that came crashing down.
President Mahinda Rajapakse ordered troops to deploy heavy equipment to speed up the rescue, his office said.
"On instructions of the president, military and army's heavy machinery have been deployed to speed up search and rescue operations," his office said in a tweet.
Military sources said they managed to move an excavator to the area and expected more heavy machinery, but damage to roads and heavy rain and mist was slowing them down.
The top military official in the area, Major General Mano Perera, said around 20 units, or about 230 soldiers, were initially deployed for the rescue in the unstable mountainous area.
Three hundred more troops were later sent, the army said in a statement.
Police commandos as well as airforce units were also deployed to dig through the mud and look for survivors.
The landslide began at about 7.45 am (0215 GMT) and lasted about 10 minutes, Perera said. "Some houses have been buried in 30 feet of mud."
Kumara of the Disaster Management Committee said hopes were fading for people who may been buried in the mud. Authorities have begun checking on the number of people who were in their homes when tragedy struck.
"We have asked local officials to check with schools and workplaces to arrive at a clearer picture of the numbers," Kumara said. "For the moment, we are maintaining the missing figure at 300."
The main focus of the search is the Meeriyabedda tea plantation, which lies close to several waterfalls.
Kumara said the mudslide struck after schools opened and tea plantation workers were supposed to be at work, but bad weather may have prompted some to stay home.
A local hospital source said two men and a woman rescued from the mud had been brought in for treatment.
Sections of several national highways have been washed away by the rains, slowing down the movement of search and rescue vehicles to the area.
A locomotive was also stuck after a mountain slope crashed onto a railway line.
The government's disaster management minister said he was on his way to the area to assess the damage.
"We are coordinating with all agencies to ensure that relief is sent as quickly as possible, but the weather is a factor that is slowing us down," the minister, Mahinda Amaraweera, told AFP.
The area is prone to mudslides and residents had been repeatedly warned to move to safer areas as monsoon rains lashed the region, the disaster management committee said.
The annual monsoon brings vital rains for irrigation and electricity generation but also causes frequent loss of life and damage to property.
Thirteen people were killed in mudslides in and around Colombo in June.
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