Abdullah was injured in the blast that occurred in the library of Imdaru Uloom madrasa in Kopaganj police station area, they said.
He was rushed to a hospital, where doctors said his condition is critical.
Abdullah's father told police that a teacher in the madrasa, Sajid, had called his son to the library for teaching purposes, they said.
The teacher is absconding. A probe has been ordered to find out how the explosives reached the library, SP Shiv Prasad Upadhyay said.