Scuffles broke out near the building, with demonstrators chanting "we want justice!" as they surged toward the doors.
After a botched snap election that lost her party its majority in Parliament, Mrs May is under fire for her response to the blaze on Wednesday.
"I know that they are looking at us to save them".
The local Grenfell Action Group had claimed, before and during a major 10-million-pound refurbishment of Grenfell Tower past year, that the block constituted a fire risk and residents had warned that access to the site for emergency vehicles was "severely restricted". Posters of missing people are pasted on many walls, and being worn as T-shirts.
But when she was asked in an interview on Newsnight on Friday night about whether she had misread the public mood, she sidestepped the question. Former cabinet minister Michael Portillo said that decision amounted to a lack of "humanity".
Mayor Khan wrote to May on Friday, saying residents felt increasingly enraged and frustrated by the slow response from the authorities.
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Hundreds of people who escaped the flames have lost their homes and all of their belongings, but Catholic parishes in the surrounding area have quickly begun receiving donations of food, clothes and water to be distributed. This will be about providing as best we possible can answers for those who have been so deeply and tragically affected by events here at Grenfell Tower.
A visit Friday to a local hospital, and the promise of a public inquiry, far from assuaging that anger.
"I called him and said: "Where are you?', I said: 'Where are you?' He said: 'I'm in the flat.' I said: 'Why you didn't come?" There has been growing fury at the low-rent estate where residents wanted answers on why the fire was able to spread so rapidly and why complaints about safety had been ignored.
"This is an absolutely terrible fire that took place". London Police's investigation remains on-going, as fire crews continue the grim and slow task of identifying and recovering bodies from inside the tower.
As many as 70 people are still missing, according to local residents and media reports, though the police haven't confirmed that number.London's fire brigade expects it to take weeks, even months, to search the block's 24 precarious floors. On Saturday, NHS England said 19 people were being treated in hospital, with 10 in critical condition.