London Fire Brigade said 10 fire engines were sent to the market, which is a popular tourist attraction.
One eyewitness said the fire was moving "very fast" and there were fears of an explosion in nearby buildings.
London Ambulance Service said it had not treated any patients but remained at the scene. The Metropolitan Police also attended the incident.
Witness Joan Ribes, 24, said: "I was just passing by when I saw the fire and they started to get firefighters and police, it was all very fast.
"We called the police to close the street to the traffic because it was very dangerous, the fire was flying through the air to the surrounding areas.
"The fire was moving very fast. People were watching, but we were scared the building could explode at any time since there are restaurants with kitchens nearby."
The ambulance service said it sent a clinical team leader and a Hazardous Area Response Team.
A police spokeswoman said: "London Fire Brigade were already at the scene when officers arrived. It is unknown at this stage if any persons are injured."