A representative of the families who met with May and spoke to reporters briefly after leaving 10th Downing Street said "we will be making this in the community, with the community".
May herself had come in for a barrage of criticism for failing to visit residents and her response to the disaster.
Prime Minister Theresa May has said the support given to survivors and the families of those who died was "not good enough" and there were "huge frustrations" on the ground as people struggled to find information.
"It is hard to escape a very sombre national mood", Elizabeth, the world's oldest and longest-reigning monarch, said in a message on her official birthday. Analysts estimate the death toll can reach 70.
"I've heard stories of heroism - from Christians, from Muslims and from others - looking after their brothers and sisters and neighbors, and doing the job that we expect from this brilliant, fantastic community that is here in this part of London", he said outside St Clement's Church, which is also operating as a relief center for survivors.
United Kingdom government must work quickly on an inquiry and consider tearing down similar tower blocks, says London mayor.
"She's a coward and needs to leave", Tilly Howard, a protester told the Guardian newspaper.
He also said: "As the mayor of London I will do my bit to be the advocate, to be the fighter, and to be the champion of these people".
Metropolitan Police Commander Stuart Cundy said the figure included the previous death toll of 30, reports the Independent. "And therefore, sadly, I have to assume that they are dead", he said.
The Wednesday blaze has so far killed 58 people while many other residents still remain unaccounted for.
Government extends interest subsidy on short-term crop loans
Interest subvention scheme will continue for one year and it will be only implemented by the Reserve Bank of India and NABARD. Under a proposal approved by the Cabinet , the Centre will provide a subsidy of up to 5% on farm loan interest, sources said.
The search of the 24-floor building had been paused on Friday due to safety concerns, but has now resumed, Cundy said. He says "my heart goes out to those affected".
Mr Cundy added: "My commitment to families is that as soon as we can, we will locate and recover their loved ones".
He said police will release images and video from inside the tower tomorrow, pending the approval of Grenfell families.
The blaze broke out in the early hours of Wednesday morning, leaving up to 58 people dead, or missing, presumed dead, and many more homeless.
In a blog post written a year ago, the Grenfell Action Group warned of "dangerous living conditions" in the tower. "It is a truly terrifying thought but the Grenfell Action Group firmly believe that only a catastrophic event will expose the ineptitude and incompetence of our landlord, the KCTMO".
But public anger has been swelling, with furious residents heckling May and storming the local authority headquarters Friday.
The leader of Britain's main opposition party is calling for the government to take over empty homes for use by residents displaced by the London high-rise fire.
Meanwhile, Queen Elizabeth II struck a mournful tone in an address to mark her official birthday Saturday, remarking not only on the blaze at the high-rise building but also on the recent terrorist attacks in Manchester and London.
The queen said in a statement that she was "profoundly struck by the immediate inclination of people throughout the country to offer comfort and support to those in desperate need". "What we are doing is putting in place the support that will help them", she added.
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