One child was hospitalized in a serious condition, the report said.
French President Emmanuel Macron has condemned the attack.
"An attack of unprecedented cowardice took place this morning in the park of Annecy. Children and adults were left between life and death. The country is in shock. Our thoughts are with the families of those attacked and those trying to save them," Macron tweeted.
Meanwhile, Agence France-Presse (AFP) reported, citing a police source, that at least six children and one adult had been injured as a result of the knife attack. Many children were about 3 years old, the report said, adding that three of them were in critical condition.
The assailant was later detained by the police, French Interior Minister Gerald Darmanin said. The man is a Syrian who has requested asylum in France, French TV reported.
Darmanin and French Prime Minister Elizabeth Borne were heading to the city of Annecy, following the incident, the French Interior Ministry said.
Later in the day, UK Foreign Secretary James Cleverly said that one of the children injured in the attack was a British citizen. He added that London had engaged UK consulate staff who were on their way to France to assist the child's family.