The fire at Grenfell Tower on the Lancaster West Estate in Latimer Road was reported at 01:16 local time.
London Fire Brigade chief Dany Cotton told reporters at the scene that there have been "a number of fatalities."
"I cannot confirm the number at this time due to the size and complexity of the building," Cotton said.
Around 250 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.
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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.
Scotland Yard has confirmed a number of fatalities and at least 50 people have been treated in hospital, according to London Ambulance.
"Extensive cordons remain in place and a number of nearby residents have been evacuated as a precaution. The A40 is closed in both directions. We kindly ask that the public stay away from the area," Cundy said.
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.
"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.
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.
London Mayor Sadiq Khan has said that 'a lot of people' were unaccounted for after tower blaze.
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.
The local Grenfell Action Grouphad claimed, before and during the refurbishment, that the block constituted a fire risk and residents had warned that access to the site for emergency vehicles was "severely restricted".
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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