The little boy was born in the central town of Brive this month and immediately bestowed with first names honouring Antoine Griezmann and Kylian Mbappe, regional newspaper La Montagne reported Wednesday.
But if the name was meant as a tribute to two of the players who helped France to World Cup glory this summer, officials were less impressed.
Authorities tasked with ensuring children's names are in their interests have flagged the case to prosecutors, the local mayor's office told the newspaper.
If prosecutors share the opinion that such an eye-catching name could cause the child problems in life, a family court could order the couple to change the name.
Such cases have repeatedly made headlines in France.
Earlier this month authorities in the eastern city of Dijon launched a legal bid to stop a mother naming her son “Jihad”.
Despite her argument that the name does not necessarily have violent connotations in Arabic — it can mean a struggle against sin within oneself — authorities worried the baby would face prejudice.