The fire at Grenfell Tower on the Lancaster West Estate in Latimer Road was reported at 01:16 local time. About 600 people are believed to have been inside the tower's 120 flats when the blaze ripped through the building.
"I can confirm six fatalities at this time but this figure is likely to rise during what will be a complex recovery operation over a number of days," said Commander Stuart Cundy, of the Metropolitan Police.
The 24-storey block, which is still on fire, looks at risk of collapsing, it said.
Around 200 firefighters, 40 fire trucks and 20 ambulance crews were at the scene at the height of the blaze.
The firefighters are still on the scene tackling the burning block and a structural engineer is on the site to ensure the building is structurally sound.
The National Health Service said a total of 74 people are being treated in hospital with 20 of those critically injured.
Officials said the block had just undergone a 10.3 million pound refurbishment.
Eyewitnesses reported seeing people trapped inside the burning building screaming for help and shouting for their children to be saved. People screamed for help as the fire took hold of the building. Some residents were seen using bedsheets to make their escape from the tower block.
As fire-fightersfought the blaze, witnesses said a baby was caught by members of the public after being dropped from the window of the 9th floor.
"This is an unprecedented incident. In my 29 years of being a firefighter, I have never ever seen anything of this scale," London Fire Brigade chief Dany Cotton told reporters.
The fire is thought to have started because of a faulty refrigerator on the 3rd or 4th floor soon after midnight and destroyed flat after flat.
However, the Met Police has said it will take some time before they are in a position to confirm the cause of the fire.
The area around Grenfell Tower is home to a large number of Muslims. Many were awake at the time the fire broke out having their early morning meal before beginning the daily fast for the holy month of Ramzan.
"I'm lucky to be alive - and lots of people have not got out of the building I've lost everything I own. I'm standing here in everything I've got," one survivor said.
The fire is believed to have broken out on the second floor of the tower block housing 120 flats over 20 residential levels and four community levels and had soon engulfed the entire building.
Eyewitnesses said they saw lights - thought to be mobile phones or torches - flashing at the top of the block of flats, and trapped residents coming to their windows - some holding children.
Another resident, Zoe, who lives on the fourth floor, said she was woken by a neighbour banging on her door.
"The whole landing was thick with smoke. The smoke alarms weren't going off but the way it spread so quickly from the fourth floor, all the way up to the 23rd floor was scary," she said.
Khan declared the fire as a "major incident" and said questions will need to be answered over the safety of such tower blocks.
"We can't have a situation where people's safety is put at risk because of bad advice being given or if it is the case, as has been alleged, of tower blocks not being properly serviced or maintained," he said.
Grenfell Tower is managed by the Kensington and Chelsea Tenant Management Organisation on behalf of the local council.
Eyewitnesses reported seeing the building burning through to "its very core".
Distraught relatives have been using social media to try and make contact with missing loved ones.
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