A man armed with a knife and an iron bar is said to have attacked people on the banks of a canal in the north-east of the city at just after 11pm local time.
Four people were injured seriously, while Le Parisien said one British tourist has a chest injury while another was stabbed in the head.
The paper said the perpetrator stabbed three people near a cinema on the Bassin de la Villette, and men playing petanque nearby attempted to stop the man, with one throwing a ball at him.
It is reported a man of Afghan nationality has been arrested and the incident is not being treated as terrorism, police said.
The Foreign Office said: "We are urgently investigating this incident and are in close contact with the French authorities."