The Metropolitan Police has confirmed "a number of fatalities", while another 50 people have been taken to hospitals across London.
Councilor Nick Paget-Brown, who represents the area in which Grenfell Tower is located, said the tower block contains some 120 individual apartments, many of which house young families, meaning the number of people trapped by the blaze could run into the hundreds.
There are no reports of Poles having been affected by a massive fire in a residential tower in west London, Polish Foreign Minister Witold Waszczykowski said on Wednesday morning.
The fire is from the second floor to the top floor of the 24 floor building.
While the cause and exact location of the fire's beginning are now unknown, some media reports have claimed the fire began around the third or fourth floor, before spreading all the way up the side of the building.
It was reported earlier that a huge fire broke out at the apartment block.
An eyewitness said people were jumping from the burning building.
The area has been cordoned off and adjacent buildings have been evacuated.
Others told the BBC that people inside the building at the time of the fire could be heard calling out for help, saying the smoke was too thick for them to leave their flats.
"This is an unprecedented incident", Fire Commissioner Dany Cotton told reporters on the scene.
The number of fatalities is not now known owing to the "size and complexity" of the building.
"It was clear that the fire service could not cope with the fire. I hope they've got everyone out". Structural engineers were monitoring the building closely, but were not immediately concerned of a potential collapse.
London Mayor Sadiq Khan said he was "truly devastated to see the horrific scenes of the major fire at GrenfellTower in Kensington".
"Our priority is to assess the level and nature of injuries and ensure those in the most need are treated first and taken to hospital", Stuart Crichton, assistant director of operations at London Ambulance Service, said in a statement.
He added: "They are being told to put a cloth over their mouths and try to get out by themselves".
Grenfell Action Group, made up of residents who live in the flats, blogged several times before about their concerns about poor fire safety.