Mexico's embassy in Russian Federation said two of its citizens were among the injured.
Citing an unnamed source, Russia's Interfax news agency reported the cab driver fell asleep at the wheel and accidentally pressed the accelerator pedal.
The extent of the injuries remains unclear although police say they are no life-threatening.
Early reports say that Mexico fans are among the injured, as some of those involved were wearing Mexican team colours. Preliminary information indicated the driver lost control of the vehicle.
Russian police said the driver in the crash has been detained and Moscow's traffic monitoring agency said the driver claimed the crash wasn't premeditated.
Eight people sought medical assistance, Moscow police said. "He was shouting: "It wasn't me".
The Moscow traffic authority has said the taxi driver said he was not acting on goal.
Asked if the driver appeared drunk, the witness said: "No, he wasn't drunk".
Pictures released by the Moscow Traffic Authority show the taxi came to a stop next to a knocked-down traffic sign.
The accident took place on Ilinka Street, approximately 650 feet from Moscow's famous Red Square and GUM shopping arcade.