The incident occurred just before the service on Sunday morning in the church of Saint-Pierre-d'Arène, south of Nice, writes Le Figaro.
According to the local newspaper Nice Matin, both a priest and a nun in the church were stabbed.
The priest received several stab wounds, but both he and the nun are said to have survived with minor injuries, writes Le Figaro.
According to the TV channel BFM TV, which refers to the police, a 31-year-old French man has been arrested after the knife attack.
French Interior Minister Gerald Darmanin also wrote on Twitter that the priest's life was not in danger and that the man had been arrested by police.
Nice mayor Christian Estrosi says in a statement according to Reuters that the suspected perpetrator is mentally disturbed and unstable.
He should not have a criminal record.
The attack took place at the same time as the French went to the polls to vote in the second, and final, round of the presidential election.