France has strict rules about the size of discounts that retailers can put on products outside of official sales periods.
Last week, shoppers at Intermarché shops jostled to buy heavily discounted chocolate and hazelnut spread.
It had been discounted by 70% - from €4.50 (£3.90) to €1.40.
Police were called when people began fighting and pushing one another.
"They are like animals. A woman had her hair pulled, an elderly lady took a box on her head, another had a bloody hand," one customer told French media.
Similar scenes were reported across France, with some being described as "riots".
France's consumer protection and anti-fraud agency said on Monday its investigation would judge whether the promotion legally counted as a sale or whether it was an instance of product dumping.
Minister of Agriculture Stéphane Travert, who is currently pushing for a law that would limit discounts to 34% of the cost price, is reported to be furious about the affair.
Nutella's makers, Ferrero, said they regretted last Thursday's violence but noted that the discount had been unilaterally decided by Intermarché.