BFM TV reported on Wednesday that an argument started over money on Tuesday lunchtime in front of the Lycée Marguerite-de-Navarre.
One of the boys took out a knife and stabbed the other at least twice.
The wounded pupil was rushed to hospital in the town where he died of his injuries later on Tuesday evening.
The alleged attacker was an 18-year-old pupil at the school who is being held by police and will appear in front of magistrates on Wednesday or Thursday.
Neither boy is thought to have been in trouble with the police in the past.
Bourges is the principal town of the Cher department, which sits in the Centre region of the country.