It is feared many are trapped inside the blazing building as the fire broke out in the wee hours of Wednesday.
In a statement on Saturday, Met Police Commander Stuart Cundy said: "Whilst I sincerely hope that our work over the coming days means that we able to say that less people are confirmed as having died, I also have to consider the sad reality that this may rise".
Cundy said there may have been people in the tower that police are not aware of, which would add to the death toll.
Cundy said police had now managed to get to the top of the tower and had undertaken a first visual search for victims, ahead of later painstaking searches.
The Metropolitan Police's investigation into the fire will look at the refurbishment undertaken at the tower block in 2016.
They also expressed anger that the prime minister did not meet with survivors or victims of the fire when she initially visited the neighborhood following the tragedy. This year, however, it is hard to escape a very sombre national mood. British health authorities say that 19 patients are still being treated at four London hospitals.
Frank adds that anger is rising in London about the deaths - for example, a leading tabloid was emblazoned with the headline "Arrest the Killers" while Labour MP David Lammy has said the incident amounts to "corporate manslaughter".More news: Trump Plans Major Restrictions on Cuba Travel
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"The Brigade advised that a section of track be temporarily closed due to the short term risk of some debris from the tower falling onto the track", a London Fire Brigade spokeswoman said. Earlier in the day, she described the country's mood as "somber" but insisted that Britain remained resolute during a hard time.
The Telegraph collected some of the known names of the dead with pictures, mostly from the accounts of people searching for lost friends and family members.
The London Evening Standard reported that hundreds of activists marched to the town hall and gathered outside the building on Friday chanting: "We want justice".
The government also faced questions about failing to apply the lessons from a similar fire in 2009. The other deceased remain inside the building.
The tragedy has provoked a huge response from nearby communities. More than 3 million pounds ($3.8 million) have been raised for the victims. Numerous displaced are living in churches and community centres. "As soon as we can, we will locate and recover loved ones", he said.
Two Underground lines near the fire area were partially shut down on Saturday to make sure that debris did not land on the tracks.
The fire forced residents to flee through black smoke down the single stairwell, jump out of windows or even drop their children from the 120-apartment building.