The six-year-old Rajonyo Sarkar, a student of South Point School, became unconscious at around 9.40 AM and was pronounced dead in a nursing home.
Guardians staged a demonstration outside the school in protest against Sarkar's death.
"A police complaint has been filed against the school authorities. We have started an investigation," a senior police official said.
The school in a statement said that the student collapsed during the break with teachers taking him to the infirmary from where he was referred to a nursing home close by where he was declared dead.
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Education minister Bratya Basu said that he had sought a report from the department secretary.
"Although the government does not have control over a private school, but cognisance can be taken if any complaint is lodged under ROPA (Revision of Pay and Allowances)," he said.
"Today this has happened in a private school, it may take place in a government school in future. The school authorities should be more careful," he said.