The tower is at least 24 storeys high and contains 120 apartments.
200 firefighters have been tackling the blaze with 40 engines.
London ambulance service said they were called at 1.29am and 20 crews raced to the scene.
London fire brigade have confirmed there have been “a number of fatalities”.
It is also confirmed that 30 patients have been taken to five London hospitals.
There have been multiple reports of people trapped in the blaze.
An eye witness at the scene said at least one man could be seen at a window waving and calling for help.
Streets around the tower have been sealed off and residents in their houses told to move out.
Falling debris also caused nearby flats to be evacuated.
London fire brigade have called it an “unprecedented incident” and say there is no known cause at the moment for the blaze.
The mayor of London, Sadiq Khan, has declared the fire a “major incident”.
Transport for London says there is no service on the Circle and Hammersmith and City lines between Hammersmith and Edgware Road.
The A40 Westway has been closed in both directions. Grenfell Tower, which was built in 1974, had recently undergone a major refurbishment.