Russian media are reporting that the fans were made up of Mexican supporters waiting to watch their team play Germany tomorrow.
The same witness said that the driver tried to flee the scene but was apprehended by police.
The taxi driver immediately got out of the vehicle and tried to run away, but people caught him.
One injured woman was in serious but not critical condition, according to Interfax, citing a medical source.
Moscow's traffic authority said the driver had a license from Kyrgyzstan, a mainly Muslim ex-Soviet republic.
The incident occurred shortly after Argentina and Iceland finished playing to a 1-1 draw in Moscow's Spartak Stadium some 13 kilometres northwest of Red Square.
The Interfax news service reports that two female Mexican citizens were among the injured.
Taxi crashed into pedestrians in Moscow.
The Kyrgyz Embassy in Moscow told RFE/RL's Kyrgyz Service that it was following the matter and that the man questioned in a police station in the city center.
However, Russia is now hosting the World Cup and as such is on high alert for potential attacks. Thousands of fans from Mexico have traveled to Moscow for the tournament, which is being held in 11 Russian cities and runs through July 15.
At least eight people sustained injuries in the incident.
The U.S. State Department on Friday updated its travel advice on Russian Federation, saying terrorist groups were plotting attacks.
The driver jumped the curb, and eight pedestrians are reported to have been injured..
"We're not scared but disappointed", said Youseff Fraige, 27, from Monterrey, Mexico.