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.
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.
Seventy-four people have received treatment in hospital. At least 20 are known to be in a critical condition.
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.
“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 said to a reporter.
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 police keep pushing back their cordons, pushing back members of the public for fear the building might collapse,” he said.