Two Mexican nationals, who had come to Russian Federation for the World Cup, were among the injured in the incident. Preliminary information indicated the driver lost control of the vehicle.
The authority cited the driver, who was in police custody, as saying he had not driven into the crowd on goal.
Russia's Interfax news agency cited a source claiming that the driver was drunk at the time of the incident.
Moscow traffic authority has published a photograph which shows two ambulances at the scene.
Moscow: At least 7 pedestrians injured by taxi near Red Square
There were no immediate details on the severity of the pedestrians' injuries, although a medical source told local media that none of them were grave.
Moscow traffic authority cited the taxi driver as saying that he did not act on goal.
The accident took place on Ilinka Street, around 200 metres from Red Square and Moscow's famous GUM shopping centre.
One witness said some of those hit were wearing Mexican team colours. He had a Kyrgyzstani driving license. Mexico play Germany at Moscow's Luzhniki stadium on Sunday.