The rain began late last week as the remnants of a typhoon fed into a seasonal rainy front, with humid, warm air from the Pacific making it still more active - a pattern similar to one that set off flooding in southwestern Japan exactly a year ago that killed dozens.
Landslide warnings are in place in over a quarter of Japan's prefectures.
Authorities have warned that landslides could strike even if the rainfall diminishes.
At least 38 people have died, four people are said to be in a serious condition and another 47 remain missing, the public broadcaster NHK reported.
'Around 1,000 people were seeking rescue by Sunday morning, but we don't yet have a complete picture of the disaster, which is enormous, ' Mutsunari Imawaka, a spokesman for the prefecture's disaster management office, said.
Thirteen were confirmed dead in Hiroshima Prefecture. Abe asked his cabinet to take "every measure to prevent the disaster from worsening by taking advance actions", Prime Minister Shinzo Abe said. Some people perched on rooftops called for help and others clung to trees while waiting to be rescued.
In western Okayama prefecture, around 200 people including children and elderly people were trapped in a hospital after a river burst its banks and flooded the surrounding area. Some automakers halted production as the rain and flooding disrupted supply chains and risked workers' safety, Kyodo news agency reported.
The JMA issued an emergency heavy rain warning in eight prefectures on July 6.
The toll in record rains that have devastated parts of Japan rose Sunday to at least 44, officials said, with authorities issuing new warnings as torrential downpours continued.
Japan Ground Self-Defense Force members use a boat to evacuate a resident from a flooded area caused by heavy rains in Kurashiki, Okayama prefecture, southwestern Japan.
Work crews could be seen elsewhere trying to clear multiple small landslides that coated roads in mud, rendering them virtually impassable. Also in Ehime, two elementary schoolgirls and their mother who got sucked into a mudslide were pulled out but could not be revived, it said.
Mr Lee said he is deeply saddened to learn that the torrential rain in western Japan over the past few days has led to the tragic loss of lives and affected many families.
One, a 95-year-old woman, was found in her house after part of a nearby mountain collapsed. Some 30,000 people remained in shelters on Sunday, while the evacuation orders and advisories were issued for almost 6 million people, according to the Fire and Disaster Management Agency.
The Hiroshima prefecture was the worst hit, with 29 people dead, while 19 were killed in Ehime prefecture, according to NHK World.