The tragedy took place at the government primary school in Dahrmasati Gandawan village at Mashrakh block, about 25 km from here, they said.
The children fell ill soon after eating the meal, which consisted of rice, pulse and soyabean, police said.
They were rushed to Chhapra sadar hospital where 11 of them were declared dead on arrival. The others are undergoing treatment, police said.
Chief Minister Nitish Kumar has ordered a high level inquiry into the incident, sources in the CMO said, adding that a Forensic Science Laboratory team will assist in the probe, which will be conducted jointly by Saran Divisional Commissioner and DIG.
Chhapra falls in the Lok Sabha constituency of RJD president Lalu Prasad.