Thunderstorms with lightning occured in several parts of the northeastern state towards the end of a day in which temperatures reached at least 37.7C, according to forecasters.
Narendra Modi, India's prime minister, offered condolences to the families of victims.
"My thoughts are with all those who lost their near and dear ones due to lightning in parts of West Bengal. May the injured recover at the earliest." the prime minister said on Twitter.
Compensation for affected families was announced by the Prime Minister's Office on Monday and will come from the National Relief Fund.
India's Meteorological Department forecasts thunderstorms in West Bengal will continue until Friday.
Lightning strikes are fairly common during India's monsoon season, which lasts throughout the summer months.
The monsoon season brings rain which is essential for the crop harvest and replenishment of water supplies across India.
But thousands are killed each year in the extreme weather across the country.
Lightning killed 2,876 people in India in 2019, more than one-third of all accidental deaths caused by forces of nature, according to the latest available figures from the National Crime Records Bureau.
Deaths by lightning accounted for more than one-third of all deaths caused by forces of nature in 2019, higher than flooding and sunstroke, the next two highest causes, put together, according to the NCRB.