In India’s Assam, at least nine people died in the floods and the homes of about two million others were submerged, the state disaster management agency said.
Lightning strikes had killed at least 21 people in Bangladesh since Friday afternoon, police officials said.
Among them were three children aged between 12 and 14, who were struck by lightning in the rural town of Nandail, local police chief Mizanur Rahman told AFP.
He said four other people were killed when landslides hit their hillside homes in the port city of Chittagong.
Both countries have asked their militaries to help amid the severe flooding, which could worsen because rains are expected to continue at the weekend.
The Brahmaputra, one of Asia’s largest rivers, breached its mud embankments, inundating 3,000 villages and farmland in 28 of Assam’s 33 districts.
“We expect moderate to heavy rainfall in several parts of Assam till Sunday. The volume of rainfall has been unprecedented,” said Sanjay O’Neil, an official at the meteorological station in Guwahati, Assam’s capital.
Several train services were cancelled in India amid incessant rains over the past five days. In southern Assam’s Haflong town, the railway station was under water and flooded rivers deposited mud and silt along the tracks.
India’s army has been asked to help other disaster response agencies rescue stranded people and provide food and essentials to those whose houses were submerged.
“We are using speedboats and inflatable rafts to rescue flood-hit people,” an army official said.
In Bangladesh, districts near the Indian border have been the most affected.
The flood forecasting and warning centre in Dhaka, the capital, said water levels in all major rivers across the country were rising. The country has about 130 rivers.
The centre said the flood situation was likely to deteriorate in the worst-hit Sunamganj and Sylhet districts in the north-eastern region as well as in Lalmonirhat, Kurigram, Nilphamari and Rangpur districts in northern Bangladesh.
Authorities sent troops to evacuate households cut off from neighbouring communities after relentless downpours over the past week, while schools have been turned into relief shelters to house entire villages inundated in a matter of hours by rivers that suddenly burst their banks.
"The whole village went under water by early Friday and we all got stranded," said Lokman, a resident of Companyganj village in Sylhet district.
"After waiting whole day on the roof of our home, a neighbour rescued us with a makeshift boat. My mother said she has never seen such floods in her entire life," he told AFP.
Asma Akter, another person rescued from the rising waters, said her family had not been able to eat for two days.
"The water rose so quickly we couldn't bring any of our things," she said. "And how can you cook anything when everything is underwater?"
Flooding in Sylhet worsened on Saturday morning after a temporary reprieve from the rains the previous afternoon, chief administrator Mosharraf Hossain told AFP.
"The situation is bad. More than four million people have been stranded by flood water," Mr Hossain said, adding that nearly the entire region was without electricity.
Flight operations at the Osmani International Airport in Sylhet have been suspended for three days as floodwaters have almost reached the runway, said Hafiz Ahmed, the airport manager.
Last month, a pre-monsoon flash flood, triggered by an onrush of water from upstream in India’s north-eastern states, hit Bangladesh’s northern and north-eastern regions, destroying crops and damaging homes and roads. The country was only starting to recover from that shock when fresh rains flooded the same areas again this week.
Bangladesh, a nation of 160 million people, is low-lying and faces threats from climate change-related natural disasters such as floods and cyclones.
About 17 per cent of people in Bangladesh will need to be relocated over the next decade or so if global warming persists at the present rate, the UN’s Intergovernmental Panel on Climate Change has said.
With reporting from agencies.