Sanat Bag (45), a labourer, was reportedly standing in queue to withdraw money for his daughter's wedding when he collapsed and died, police said.
Two of his sons are mentally challenged.
Manager of the bank, Uday Bhanu Pal, said he has heard that the man died outside the bank building.
An inquiry has been launched into the incident, police said.
Trinamool MLA from Uluberia South Pulok Ray visited the spot and reported the incident to Chief Minister Mamata Banerjee.
However even after the deadline ended, the man died standing in a bank queue today and the Centre should be ashamed for it, he said.
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