Weakened by a botched election gamble last week, May has been criticised for her muted response to the fire and had to be rushed away from a meeting with residents on Friday under heavy police guard as protesters shouted "Shame on you".
The fire at the 24-story building has led to community anger and protests over the British government's response.
As police continued their investigation into the inferno that killed at least 58 people, Trade Minister Greg Hands said the Government was carrying out an "urgent inspection" of the roughly 2500 similar tower blocks across Britain to assess their safety.
While the blaze has prompted an outpouring of generosity, with many people donating provisions and clothes, it has also unleashed rage at the authorities as the charred tower was cast as a deadly symbol of a deeply divided society.
"We do not want another fatality arising out of this tragedy". Residents want answers on why the fire spread so quickly, trapping numerous estimated 600 residents.
She has also pledged £5 million (S$8.8 million) of support, housing guarantees and help with access to bank accounts and cash.
"The residents of Grenfell Tower, families who have lost loved ones, and the emergency services who have been working so hard to help them have been through some of the most harrowing and traumatic experiences imaginable", Mrs May said.
Cundy said the images and video of the scenes inside the Grenfell Tower would be released on Sunday, adding "We will not do that until we have contacted all the families we are supporting to let them know that's our intention".
58 missing and presumed dead in London apartment fire
United Kingdom government must work quickly on an inquiry and consider tearing down similar tower blocks, says London mayor. A resident in a nearby building watches smoke rise from a building on fire in London , Wednesday, June 14, 2017.
The first secretary of State in her Cabinet, the equivalent of deputy PM, Damien Green came to her defence. We are all desperately sad, we are angry, but of course none of us as angry as those who were directly affected. "We're all angry, but of course none of us as angry as those who were directly affected".
"I spoke to the DPP yesterday and there are prosecutors already in advising the police", said Starmer.
Nick Paget-Brown, the Tory leader of Kensington and Chelsea council, said there were officials on the ground "very soon" after the fire broke out to help residents, but he sidestepped questions over whether he feels guilty about what happened. "We'll get to the bottom of this", she said.
May is chairing a government task force on the fire and a spokesman says that she will meet afterward with "a group of residents, victims, volunteers and community leaders" at No. 10 Downing Street. She also says she will receive daily reports from the stricken neighborhood.
"It is hard to escape a very sombre national mood", Elizabeth said in a message on her official birthday.
"Put to the test, the United Kingdom has been resolute in the face of adversity", Elizabeth said.
Opponents said Mrs May's handling of the fire has thrust her position further into doubt.
John McDonnell, the shadow chancellor, criticized the Government's response and a Labour MP, David Lammy, raised fears of a cover-up as he urged the Prime Minister and Metropolitan Police to immediately seize all relevant documents. "As prime minister, I will be responsible for implementing its findings".