Police in Moscow said eight people were injured when a taxi crashed into pedestrians on a pavement near Red Square.
It's not clear how badly injured the victims were, TASS said.
The witness added some of those hit were wearing Mexican team colours.
The Russian home secretary told the press that the driver lost control of the auto, despite video footage seeming to suggest that it was intentional.
Police said seven people were hurt in the incident, which took place as residents and visiting soccer fans thronged the centre of Moscow on a balmy summer evening. Mexico kick-off their campaign at Moscow's Luzhniki stadium and thousands of their fans are in the Russian capital.
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The ICO said it was liaising with the National Cyber Security Centre, the FCA and other agencies to determine the impact on customers.
Moscow police said the driver has been detained and a criminal investigation into a suspected violation of the traffic code has been opened. He was shouting: "'It wasn't me'".
The driver, identified by Moscow city authorities as a national of the central Asian republic of Kyrgyzstan, was taken into custody for questioning.
Interfax news agency cited a source as saying that the driver was drunk at the time of the incident, which occurred near Red Square.
Moscow is one of 11 Russian cities where tournament games are being played.