Social distancing was futile for one frightened group of people fleeing Cyclone Amphan, who herded cows and chickens to a packed Indian evacuation bunker on Wednesday despite fears of coronavirus infection.

The shelter, west of Kolkata in Midnapore district, opened just before the strongest storm in decades collided into the nearby coast packing winds of up to 190 kilometres per hour (118 mph).

Twenty people were cramped in one small room and only two were wearing facemasks, even though authorities had pledged to reduce crowding in shelters and make the wearing of protective gear compulsory.

"I don't think I have a mask," said Kavita Lahiri, who brought her three children to the concrete building but had to leave her two cows tethered outside.

Amphan is set to cut a devastating path through eastern India and neighbouring Bangladesh at a time when both countries are struggling to contain coronavirus outbreaks.

At least 650,000 people fled to evacuation shelters in India's West Bengal and Odisha states and an estimated 2.4 million were relocated on other side of the border ahead of the cyclone's landfall.

Authorities fear that migrant workers who recently returned from major cities could be carrying the virus to impoverished rural districts around the Bay of Bengal that now lie in the storm's path.

"Cyclone Amphan is the perfect example of how interconnected our crises are -- with the poor having to cram into crowded cyclone shelters and put themselves at risk of catching the virus," said leading Bangladesh social activist Risalat Khan.

The West Bengal government said it had sent masks and sanitiser to evacuation centres, but at most shelters there was little sign of protective equipment.

Across the border in Bangladesh's Khulna district, more than 200 anxious villagers packed the Momtaj Begum school.

"We are worried because of the cyclone and the coronavirus," said 25-year-old Rumki Khatun as she cradled her infant son.

"The room is already packed and maintaining social distancing is impossible here. Only Allah can save us."

Bangladesh's junior disaster management minister Enamur Rahman told AFP that Bangladesh had tripled its number of evacuation shelters to nearly 15,000 to help social distancing.

"People have been asked to wear masks. We have also made provision for soap and sanitiser," he added.

But many others in the storm's path said they would not leave their homes out of fear of catching the virus.

"We heard that the cyclone shelter near the police station is crammed with people," said Sulata Munda, a mother of four in nearby Shyamnagar district.

Her family and neighbours had all stayed behind as well.

"The village guard told us to leave. We fear the cyclone, but we also fear the coronavirus. Many of us did not go," she said.

The virus has claimed more than 3,000 lives in India and nearly 400 others in Bangladesh, according to official figures.

Experts say the low level of testing in both countries means the true toll is likely higher.