Deborah Loupien-Suares, the head of the Jewish community in the towns of Bayonne and Biarritz, said she discovered the damage on Sunday when visiting the graves of her grandparents.
“There is significant damage to up to 10 tombs at the cemetery which have been smashed,” she told AFP, expressing her “shock and horror”.
Tombstones had been broken as well as a commemorative plaque for a girl who was deported during World War II.
Loupien-Suares said she would file a criminal complaint with police in Bayonne, where prosecutors confirmed an investigation was underway.
“There is no anti-Semitic graffiti and I don't want to inflame a debate. I want the investigation to take place calmly,” she said.
But she added that the Catholic cemetery which “is situated just opposite and is more easily accessible” was not damaged.
Bayonne Mayor Jean-Rene Etchegaray visited the scene on Sunday. The Jewish cemetery was created in the late 17th century and progressively extended.
“This is the first time this has happened in Bayonne, where the Jewish community has been perfectly integrated for years,” Loupien-Suares said.
Two Jewish cemeteries have been attacked in recent months in Alsace, eastern France.
Over 100 graves were defaced with swastikas and other anti-Semitic graffiti at the cemetery in Westhoffen in December, while 96 tombs were desecrated at Quatzenheim, also in Alsace, in February.
The rising number of anti-Jewish offences reported to police – up 74 percent in 2018 from the previous year – has caused alarm in a country that is home to both the biggest Jewish and Muslim communities in Europe.
President Emmanuel Macron said after the Westhoffen vandalism that “Jews are and make France” and that “those who attack them, even their graves, are not worthy of the idea we have of France.”