Four fire engines and around 25 firefighters are attending the blaze on a balcony on the sixth floor of the building on Bridges Court Road in Wandsworth. The London Fire Brigade was alerted to the incident just after 12.30pm.
The cause of the blaze is not yet known.
In a statement, the London Fire Brigade said: “Four fire engines and around 25 firefighters are tackling a balcony fire at a block of flats on Bridges Court Road in Wandsworth.
“Part of the balcony on the sixth floor is alight.
“The Brigade was called at 12.33pm.
“The cause of the fire is not known at this stage.”
One witness took to Twitter to post a video of the blaze at the block of flats.
The clip shows smoke rising from a building.
The Twitter user with the handle @bobpadron said: “Fire in Wandsworth. Hope everyone is okay!”
It comes as the UK is facing travel disruption, closed schools and health warnings as the country braces for extreme heat over the next two days.
The mercury is set to soar into the high 30s, and possibly past a sweltering 40C as the heatwave sweeps the country.
The first ever red extreme heat warning has been issued by the Met Office across a swathe of England for Monday and Tuesday.
London is predicted to be hotter than the Caribbean, the Western Sahara and popular holiday destinations in Europe.
The chief executive of the Met Office confirmed Britain “may well see the hottest day in history” today.
And Tuesday is expected to be even hotter, with some forecasts estimating highs of 43C (109F).