Eyewitnesses described people trapped in the burning Grenfell Tower, in north Kensington, screaming for help and yelling for their children to be saved.
Firefighters rescued "large numbers", but London Mayor Sadiq Khan said "a lot" of people were unaccounted for.
The 24-storey block, which is still on fire, looks at risk of collapsing.
During the night, 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.
It is understood that "several hundred" people would have been in the block when the fire broke out shortly after midnight, most of them sleeping.
Media captionMickey, a resident of Grenfell Tower: 'It was like a horror movie'
Commander Stuart Cundy, of the Metropolitan Police said: "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."
He said it was likely to be some time before police could identify the victims, adding that it was too early to speculate on the cause of the fire.
An emergency number has been set up on 0800 0961 233 for anyone concerned about friends or family.
More than 50 people are being treated in hospital, London Ambulance Service says.
By mid-morning, the building looked to be just smoking ruins but the fire has again taken hold, and cladding is falling to the ground.
Paul Munakr, who lives on the seventh floor, managed to escape.
"As I was going down the stairs, there were firefighters, truly amazing firefighters that were actually going upstairs, to the fire, trying to get as many people out the building as possible," he told the BBC.
He said he was alerted to the fire not by fire alarms but by people on the street below, shouting "don't jump, don't jump".
Eyewitness Jody Martin said: "I watched one person falling out, I watched another woman holding her baby out the window... hearing screams.
"I was yelling at everyone to get down and they were saying 'We can't leave our apartments, the smoke is too bad on the corridors.'"
Michael Paramasivan, who lives on the seventh floor with his girlfriend and young daughter, said he ignored official advice to stay in your home.
"If we had stayed in that flat, we would've perished. My gut instinct told me just to get the girls out. I wrapped the little one up because of the smoke and I just got them out."
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."
Grenfell Tower, built in 1974, is part of the Lancaster West Estate, a sprawling inner-city social housing complex of nearly 1,000 homes.
The BBC's Andy Moore, who was at the scene through the night, described watching debris falling from the building, and hearing explosions and breaking glass.
"The police keep pushing back their cordons, pushing back members of the public for fear the building might collapse," he said.
London Fire Commissioner Dany Cotton said it was too early to speculate on the building, although it was structurally safe enough for her crews to be working inside.
She urged all residents to make sure they had reported themselves to police so that the authorities know they are safe.
Mayor Sadiq Khan said he was devastated by the horrific scenes, attended by more than 250 firefighters and 100 ambulance medics.
Questions will need to be answered over the safety of tower blocks, he told BBC Radio.
"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.
Matt Wrack, of the Fire Brigades Union said something had clearly gone badly wrong with fire prevention procedures at the building.
Firefighters would normally fight a fire in a tower block from the inside, going up the fire escape, and fighting using the internal dry-rising mains, he said, but that's not been possible in this case.
Appeals are being made on social media for news of missing friends and relatives, who might have been caught in the blaze.
Emergency rest centres have opened for those now homeless at Latymer Community Centre, St Clement's Church, Harrow Club and Rugby Portobello Trust. There are also local collections under way for spare clothes, toys, blankets and toiletries.
Grenfell Tower underwent a two-year £10m refurbishment as part of a wider transformation of the estate, that was completed last year.
Work included new exterior cladding and a communal heating system.
The 24-storey tower, containing about 120 flats, is managed by the Kensington and Chelsea Tenant Management Organisation (KCTMO) on behalf of the council.
The local Grenfell Action Group had claimed, before and during the refurbishment, the block constituted a fire risk and residents had warned that access to the site for emergency vehicles was "severely restricted".
Council leader Nick Paget-Brown said the buildings were regularly inspected, but a "thorough investigation" was needed.
