The building itself contains 120 apartments, with roughly 600 residents. Firefighters have been pulled out of the tower.
There were reports that some leapt out of windows.
The London Fire Brigade tweeted that the fire involved the second to the top floor of the 27-story building.
The group said there was one entry and exit to Grenfell Tower during improvement works at the block on Latimer Road and it had issues with evacuation procedures at the building.
The London Ambulance Service said earlier that it sent "a number of resources".
Residents said they had warned repeatedly over fire safety in the block. "Those who escaped the blaze were the ones who both woke up and ignored that advice".
One man broke down in tears as he told of seeing parents throw their children from the building in an effort to save them.
"I sure would have been dead if I weren't awake", he said.
Why Britain Still Doesn't Have a New Government Confirmed
But Ruth Davidson, the pro-EU leader of the Conservatives in Scotland , called on May to "reopen" the government's Brexit plans. The talks revolve around support from the DUP on a vote-by-vote basis in parliament, rather than a formal coalition government.
One of the two-bed flats, on the 18th floor of the building, was listed to rent at the end of the a year ago for £1,971 a month. It was as silent as it could be. An emergency number has also been established for friends and family who are anxious for loved ones, known to be residents in the building.
Desperately trying to avoid the flames, residents threw one baby and other small children from high windows to people down on the sidewalk, witnesses said.
According to local officials posting on social media, 200 firefighters are attempting to put out the blaze which was reported after 1 a.m. local time.
A witness said she saw the baby dropped from the ninth or 10th floor of Grenfell Tower in London. "Somebody did, a gentleman ran forward and managed to grab the baby".
Lamrani said: "People were starting to appear at the windows, frantically banging and screaming".
"This is an unprecedented incident", Fire Commissioner Dany Cotton told reporters.
On June 14, just after midnight, a fire began on the fourth floor of Grenfell Tower located in north Kensington, a district of west London. Volunteers were bringing sandwiches to people forced out of their homes; others donated supplies and money.
The Grenfell Action Group said it had "posted numerous warnings in recent years about the very poor fire safety standards at Grenfell Tower and elsewhere in RBKC", referring to the Royal Borough of Kensington and Chelsea, the local authority which owns the tower.
He said: "The fire started happening on third floor, we called the fire bridge who came 20 minutes later, and then the whole thing just went up".
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