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.