A driver shouted "Allahu Akbar" as he ploughed into groups of people in the eastern French city of Dijon on Sunday evening, AFP reports.

11 people were injured, two of them seriously, in the attack, a source close to the investigation said.

"The man, born in 1974, is apparently imbalanced and had been in a psychiatric hospital," the source told AFP, adding that "for now his motives are still unclear".

The driver was arrested by police after targeting passersby at five different places in the city on Sunday evening, a police source said.

"Nine people were lightly injured and two others seriously but their lives do not appear to be in danger," the source said.

Witnesses told police that the driver shouted "Allahu Akbar" and "that he was acting for the children of Palestine," a source close to the investigation said.

The attack in Dijon comes the day after a French convert to Islam was shot dead after stabbing three police officers while also reportedly crying "Allahu Akbar" in the central town of Joue-les-Tours.

The perpetrator was a French citizen born in Burundi in 1994, a source told AFP, and was known to local police. He was allegedly "known for his extremist views."

His brother was known to security agencies for his radical beliefs as well, according to the International Business Times, and had at one point planned to travel to Syria.

Security was stepped up at police and fire stations across France on Sunday after Saturday’s attack.