Metropolitan Police Commander Stuart Cundy warned that the death toll could rise further as he formally identified a first victim as 23-year-old Mohammed Alhajali.
Earlier today, Prime Minister Theresa May met victims of the blaze at Downing Street, amid criticism she had not seen them in the immediate aftermath of the tragedy.
Mr Cundy said: “Sadly, at this time there are 58 people who we have been told were in the Grenfell Tower on the night that are missing, and therefore sadly, I have to assume that they are dead.
“That number 58 may change. I really hope it won’t, but it may increase.
“Our focus has been on those that we know were in Grenfell Tower. However, there may be other people who were in there on the night that others were not aware were there.
“That is also an absolute priority for the investigation – to establish who they may be,” he added.
Mr Cundy also appealed to anyone who may have escaped from the building, but has not yet come forward, to make themselves known.
Of the 58, he said 30 were confirmed dead. A total of 16 bodies have been recovered and taken to a mortuary.
Mr Cundy said the police investigation into the blaze would look at the building and its refurbishment in 2016 and vowed to prosecute people “if there is evidence”.
He said: “The investigation is a police investigation. We investigate criminal matters. The investigation will identify any criminal offence that has been committed. It will be wide ranging.
“It will go to establish the answers of what happened in the fire and how it spread, it will look at the building itself, it will look at the refurbishment as well.
“Our criminal investigation will identify any criminal offences that have been committed.
“Wherever we can, we will bring people to justice if there is evidence. It is completely and wholly inappropriate for me to talk about details of the investigation which may subsequently jeopardise any criminal proceedings.”
Mr Cundy added: “The time it is going to take us to undertake the search and recovery operation is going to be significant.
“Both myself and colleagues from London Fire Brigade have already said it will take weeks. It may take longer than that.
“My commitment to families is that as soon as we can, we will locate and recover their loved ones.
“The reason we had to pause the search and recovery yesterday was for the safety of our staff.
“We do not want another fatality arising out of this tragedy.”
Mr Cundy’s statement came as Transport for London partially closed two Tube lines after fire chiefs said there was a “short-term risk of some debris falling on to the tracks”.
When asked about the search, he replied that the officers had gone “all the way to the top” of the tower, and explained that the first phase was a visual check.
“There is considerable damage within Grenfell Tower. It is important for families that we do absolutely everything in there to find their loved ones but we have gone to the top of the tower,” he said.
Family liaison officers are working with 52 families, and as soon as victims are identified, their loved ones will be told.
Mr Cundy said at the scene: “At this point in time we have 16 people who have been recovered to the mortuary.
“I absolutely understand the frustration of why figures haven’t been released earlier.
“The reason for that – at one point, in terms of our casualty bureau, there were 400 people who were reported missing from Grenfell Tower.
“Grenfell Tower itself is 120 flats. We have worked tirelessly over the last four days to truly understand those that we know were there on the night.”
He added: “I understand – I really do understand – the frustration of so many about not knowing the scale of the tragedy that is unfolding behind us.
“I have said it before, you have my absolute assurance that as soon as I can possibly tell you something that I know to be accurate, I will tell you.”
Anger flared in the Kensington community over the weekend, as some accused the authorities of withholding information.
Mr Cundy said: “The investigation will be exhaustive. My intention is that it will help provide answers.
“If, as we investigate, we identify issues that are a risk to public safety, we will not be waiting until the end of the investigation before we provide that information to the appropriate authorities.
“If there are any safety issues that we and experts that we will be using identify, we will share that immediately.”
The police are appealing for anyone with pictures or videos of the blaze to hand them in, as they may help establish not only where and how the fire started, but also how it spread.
Meanwhile, victims made clear their demands to the Prime Minister in a two-and-a-half hour meeting in Number 10.
A man representing the group, who did not give his name, told reporters they would make a full statement “in the community”.
He said the group had spoken about their “demands and what we expect”.