The 11 are said to include a man, his two sisters and mother, along with their respective spouses and children, including a baby.
The family is believed to have left at the end of September, the source said yesterday.
The wife of the man was identified by the Nice-Matin newspaper as 27-year-old Andrea, who left with her two sons aged four and six.
"All is well. I am in a superb region," she told her parents, according to their account in the paper.
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Her father told the paper she had converted to Islam.
"I saw how religion played a bigger and bigger part in her life," he said. "Perhaps I should have reacted."
The Muslim father of Andrea's husband, Reda, confirmed his entire family had disappeared, including his wife, twin daughters and four grandchildren.
"We were married since 1983," he told Nice-Matin.
"I knew religion held an important place for them, but only up to a point. My whole family has disappeared, and I'm scared."
France has Europe's biggest Muslim population and is thought to have provided the largest contingent of Western jihadists to the conflict in Syria.
More than 70 similar investigations have been launched since the start of the year.
Prime Minister Manuel Valls told MPs last month that around 1,000 French citizens are involved in jihadist networks, with an estimated 580 having travelled to fight in Syria and Iraq.