Police say 58 people at the tower are now confirmed or presumed dead.
Commander Stuart Cundy said he expects the ongoing search and recovery effort to last "many, many weeks", as authorities work to identify the victims.
Five people have been formally identified after the blaze tore through the residential block last Wednesday in west London.
The building had recently underwent an $11 million upgrade, triggering questions about how it became a blazing concrete coffin.
Concerns have been raised about the cladding installed on Grenfell Tower.
"I do think there may be some change to that number", Cundy told reporters.
"I have always said I will be accurate about what I know, so the next figure of those presumed dead and missing will be released tomorrow, Monday 19 June". The public is also demanding answers about how the blaze spread so quickly amid reports that the recently-renovated building's exterior paneling fueled the flames.
Shortly after Saturday's meeting, Downing Street released a strongly-worded statement in which the Prime Minister said support in the immediate aftermath of the fire "was not good enough".
Mrs May said: "I can confirm that a £5m emergency fund that I announced yesterday is now being distributed on the ground so people can buy clothes, food and other essentials".
Fire safety experts say despite that outcome, "stay put" is still the best advice if fire breaks out in a different part of a high rise building.More news: Britain, EU start historic Brexit talks
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Meanwhile, according to the Associated Press, Trade Minister Greg Hands said Sunday that the government is carrying out an "urgent inspection" of the roughly 2,500 similar tower blocks across Britain to assess their safety, while an opposition lawmaker urged the government to quickly secure documents in the Grenfell renovation for the criminal investigation.
Speaking to the Observer, she said crews did not follow normal practice, adding: "Had we just followed standard fire brigade procedures, we would not have been able to commit firefighters in and conduct the rescues we did". She said many identifications would probably be done via dental records, predicting that such samples would be more likely found from people who died of smoke inhalation, rather than those killed by the fire itself.
"I'd like to reassure everybody that we will be looking at all criminal offenses that may have been committed by any individual or any organization", he said.
Cdr Cundy said many families lost more than one relative in the horror blaze.
NHS England says the injured are being treated in four London hospitals.
London Mayor Sadiq Khan has attended a church service near the ruined London high-rise apartment building where at least 58 people perished.
Queen Elizabeth II marked her official birthday Saturday by saying Britain remains "resolute in the face of adversity" after the horrendous fire and recent extremist attacks in London and Manchester.
On Friday angry protesters chanting "We want justice" stormed their way into the Kensington and Chelsea town hall to try to confront the leaders of the local council.
"It is hard to escape a very sombre national mood", Elizabeth said in a message on her official birthday.
The fire and rescue service cited "the short term risk of debris falling onto the track".