The driver, a citizen of Kyrgyzstan, said he did not act on goal, according to Moscow authorities.
A witness at the scene told Reuters that some of the people hit were wearing Mexican team colors.
Eight injured after taxi ploughs into ‘World Cup fans’ in Moscow
They added that the taxi driver likely lost control of the vehicle.
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The driver is reported to have tried to flee the scene, but was arrested by police.
Russia's Interfax news agency cited a source claiming that the driver was drunk at the time of the incident.
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Pictures released by the Moscow Traffic Authority show the taxi came to a stop next to a knocked-down traffic sign.
The city health department said seven of the injured were in satisfactory condition and one was in more serious condition.
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The accident took place on Ilinka Street, about 200 metres from Red Square and Moscow's famous Gum shopping arcade.