The city's mayor said the driver lost control of the vehicle, while Russia's Interfax news agency quoted an unnamed source as saying he had fallen asleep at the wheel.
The driver in the crash on Saturday has been detained, police said.
The Moscow Traffic Authority says the taxi driver has a Kyrgyzstan driving licence.
A witness at the scene told Reuters that some of the people hit were wearing Mexican team colours.
Russia is now hosting the soccer tournament and Moscow is one of 11 Russian cities where World Cup games are being played. He was shouting: "'It wasn't me'". There are no fatalities but at least one woman has serious injuries although she is not believed to be critical, according to Interfax.
The accident took place on Ilinka Street, approximately 650 feet from Moscow's famous Red Square and GUM shopping arcade.
Russian Federation hosts the 2018 FIFA World Cup from June 14 to July 15.