He said: "The building itself is in a very hazardous state".
The block, which housed some 600 people in about 120 apartments, is in a low-rent housing estate next to one of the most affluent areas of Britain.
"I have friends in the tower and they are not telling us anything", said Salwa Buamani, 25, who came with her three-year-old niece on her shoulders.
She was speaking as she took her place as Honorary judge in the annual competition that works to encourage literacy and a love of writing and reading stories.
Meanwhile, Queen Elizabeth II and her grandson Prince William visited a local sports center where Grenfell residents who lost their homes are being temporarily accommodated. They cited a series of unanswered issues including whether the cladding used on the building helped the blaze spread.
Told there was a need for the public to hear her say something had gone badly wrong and the Government accepted responsibility, Mrs May said: "Something awful has happened". Six victims have been identified and the others have been located, but with dozens of people still apparently unaccounted for speculation has increased that the number of dead could hit three figures. However, besides allegedly using cheaper less fire resistant panels, The Guardian also discovered a long and twisting list of companies and subcontractors involved in the refurbishment of the tower.
In addition, London Mayor Sadiq Khan called for an interim report on the fire to be published this summer. Former cabinet minister Michael Portillo said that decision amounted to a lack of "humanity".
"I do sense the anger", Conservative lawmaker Andrea Leadsom said as she faced the fury of local residents during a tour of the site. "Sadly, it is expected that the total will rise and it is not expected that any survivors will be found".
The local Grenfell Action Group had claimed, before and during a major 10-million-pound refurbishment of Grenfell Tower previous year, that the block constituted a fire risk and residents had warned that access to the site for emergency vehicles was "severely restricted".
"What she wanted to do was to speak to the people working on the ground on the recovery operation, the rescue operation to make sure that they've got everything they want and see how she could help", he told Sky news.
He said: "I'm not here because I support the Tories, Labour or Liberals".
Police said on Thursday that they had launched a criminal investigation into the fire.
The government said checks were now planned on tower blocks that have gone through a similar upgrade.
This number is likely to rise further, Metropolitan Police Commander Stuart Cundy has warned.
She was forced to answer questions over why the Prime Minister hadn't met survivors during her earlier visit.
Twenty firefighters and four fire engines remained at Grenfell Tower throughout last night (local time), the London Fire Brigade said.