There is now no details on the condition of those who were injured. Police said the driver in the crash was detained, but other Russian authorities said he had stated the crash was accidental.
An unidentified official said of the injured: "Seven of them are in a satisfactory condition and one woman is in a moderate condition".
Russia is now hosting the World Cup and Moscow is one of 11 Russian cities where tournament games are being played.
"In relation to the incident in the centre of Moscow today, we inform that two Mexican citizens have slight injuries and their health status is stable and not at risk", the Mexican embassy posted on Twitter. He was shouting: "'It wasn't me'".
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Jose Gimenez came to Uruguay's rescue by scoring the lone goal in a 1-0 victory over Egypt on Friday in Group A of the World Cup. Cuper is known to be a conservative manager, but he may need to crank up the offense if Suarez and Cavani catch them early.
Moscow's traffic management authority said the taxi driver had a driver's licence issued in Kyrgyzstan, a mainly Muslim ex-Soviet republic.
Two ambulances quickly arrived at the scene and seven people were taken to hospital.
Asked if the driver appeared drunk, the witness said: "No, he wasn't drunk".
The incident took place on Ilinka Street, about 200 meters (650 feet) from Red Square and Moscow's famous GUM shopping arcade.