Costa was taken to a hospital and received nine stitches before being released.
Police say they have no report of the case – and that police don’t patrol inside the Fan Fest organized by the city government.
Rio’s mayor’s office didn’t immediately reply to an email requesting information.
Earlier on Saturday, a report surfaced that an England fan had his ear bitten off by another after the national side’s World Cup game against Uruguay. The man was left with half his left ear ripped off and bloodied in the incident, the Association of Chief Police Officers confirmed.
The incident happened before Uruguay and Liverpool striker Luis Suarez was banned from the competition for biting an Italian player but details of the attack have only just emerged.