Nearly all of the building had been searched but crews were still trying to put out "pockets of fire" in hard to reach places and "unknown numbers" remain inside, London Fire Brigade Assistant Commissioner Steve Apter said.
Scotland Yard said the death toll of 17 is expected to rise further as the emergency services have lost hope of rescuing any more people alive from the building.
"Sadly I can confirm that the number of people who have died is now 17", London police commander Stuart Cundy told reporters.
Cundy said identifying victims will be very challenging and police will use global disaster victim protocols which include dental records, fingerprints and DNA.
London police said an investigation, led by a detective from its homicide and major crime unit, would examine whether criminal offenses had been committed although they said there was nothing to suggest the fire was started deliberately.
The probe was launched when it emerged that cladding panels similar to those believed used in Grenfell Tower have been banned on tall buildings in the US.
Mrs May, who made a brief, private visit to the scene earlier, said: "The emergency services told me that the way this fire spread and took hold of the building was rapid, it was ferocious, it was unexpected".
Residents organisation Grenfell Action Group had earlier claimed that the block constituted a fire risk.
One person said: "You know this could have been stopped right?"
The many questions being asked now include whether a "stay put" fire protocol, which called for residents to remain in their apartments if there was a fire elsewhere in the building, might have turned a lethal fire even more deadly. "We need to know exactly the number of people who were there during this tragedy".
Queen Elizabeth II and her grandson Prince William visited a community centre where some of the survivors are being housed, as anger grows among local residents about allegations that fire safety concerns were ignored for years.
"We entirely support the calling of the public inquiry and will cooperate in whatever way we can with it so that local people have all the answers about what has happened", the council, the Royal Borough of Kensington and Chelsea, said.
"I've seen a lot of bad things and not far away from where I'm sitting, it really dawned on you".
One of the victims was named as Mohammed Alhajali, a 23-year-old Syrian refugee, who came to Britain in 2014 with his brother.
One of 30 known victims died in hospital. "And once it's safe, they are going to go into the building", he said, in an attempt to calm the crowds.
Grenfell Tower is part of a social housing estate in north Kensington, just streets away from some of the most expensive homes in the world in Notting Hill.