Karanbir Cheema was taken to a west London hospital after becoming unwell during a school break.
The head of the school in Greenford said Karanbir was treated immediately in the school before paramedics arrived.
He died on Sunday after 10 days in the Great Ormond Street children's hospital, the reports said.
Officers from Ealing borough police were initially called to the school after it "became apparent that an incident had occurred which led to the boy coming into contact with the allergen," the Metropolitan police said in a statement.
Karanbir's father, Amarjeet Cheema, told the Evening Standard how he had watched his "brilliant" son die in hospital.
"While he was in hospital we were fully concentrated on his condition. Now we want answers. How could this have happened?" Mr Amarjeet said.
A statement on the school's website said: "The school community is deeply saddened at the loss of a Year 8 student who died on Sunday July 9 at hospital following a severe allergic reaction.
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