"Whilst I sincerely hope that our work over the coming days means that we (will be) able to say that less people are confirmed as having died, I also have to consider the sad reality that this may rise", Metropolitan Police Cmdr.
Protesters in London accused authorities of suppressing the true death toll, an idea that has spread on social media.
"I have heard the concerns and I have ordered immediate action across the board to help victims' relatives and the survivors", she said.
After meeting with Grenfell survivors on Friday, Ms May announced a 5 million pound (AUS$8 million) fund to help them and expressed sorrow for their plight. "Phone lines will have more staff". They would wear high-visibility clothing so they could easily be found, dispense advice and ensure the right support is provided, she added.
The mayor of London Sadiq Khan has said the Grenfell Tower disaster was a effect of years of neglect by politicians.
The 23-year-old Alhajali is the only victim of the Grenfell Tower fire to be officially named as the hard process of identifying human remains continues.
The identification of the victims is proving very hard - which experts attribute to the extreme heat of the fire.
He said the government is carrying out an "urgent inspection" of other tower blocks in Britain to assess safety.
There is simmering anger in the multi-ethnic north Kensington area hit by the blaze, and public fury has been directed at senior government figures, including May, who was jeered Friday after she visited. This year, however, it is hard to escape a very sombre national mood.More news: Rubio: Trump firing Mueller 'not going to happen'
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Public anger is mounting as residents and neighbors demand answers for how the blaze early Wednesday spread so quickly and trapped so numerous tower's 600-odd residents.
Protesters sit down in Oxford Circus during a rally calling for justice for those affected by the Grenfell Tower fire on June 16.
"I spent several hours yesterday talking to local residents around Grenfell Tower".
The tragedy has provoked a huge response from nearby communities. In recent years, London's skyrocketing home prices and housing shortage have left residents struggling.
Many survivors are sleeping on the floor in community centers and there's still no coordinated distribution of donated food and clothing.
The 1974 concrete building had recently been fitted with new insulation cladding.
The images of the 24-storey building being ravaged by fire "should be forever seared into our nation's collective memory", he said.