The emergency service said it had also sent 10 fire engines to the scene, near a nightclub and a covered market.
There were no reports of any casualties.
"The first, second and third floors, plus the roof, of a building within the market are alight," the London Fire Brigade said on Twitter.
A London Ambulance Service spokeswoman said they were called to a fire at 00:07 am (2307 GMT on Sunday).
"We sent a clinical team leader and our Hazardous Area Response Team to the scene.
"We have not treated any patients, and we remain at the scene," she said.