Police Commander Stuart Cundy said police will seek criminal prosecutions if the evidence warrants.
He said the important thing is to find answers for the families that have been affected.
The statement officially confirmed that a 23-year-old Syrian refugee was the first victim of the huge fire.
'But being with colleagues from the London Fire Brigade when I was in there, colleagues from the London Ambulance Service and other police officers, I think it's fair to say it is incredibly emotional working in there.
May, whose political position was already weakened by a poor showing in parliamentary elections last week, announced the government would immediately provide 5 million pounds ($6.39 million, 5.7 million euros) to help victims of the fire.
The local authority and the government have been criticised for their response to the emergency by those who have lost their homes because of the fire and by community leaders, and politicians.
Over the last four days, British police have worked hard to try to provide greater clarity on the number of people that are believed to have died in the fire.
The death toll had previously stood at 30 and officials added that it would likely take weeks for all the bodies to be recovered and some may never be identified.More news: Apple, Ikea to Launch Virtual Furniture App
An inquiry should be properly resourced so it can get the full truth about the fire as quickly as possible, he said.
Khan described the fire as a "preventable accident that didn't need to happen", adding that the "tragedy we're seeing is a outcome of the mistakes and neglect from politicians from the council and from the government".
"We must prepare people for the bad reality that some people may not be identified due to the intensity of the fire", he said.
He urged those people to make themselves known to the authorities.
They demanded justice for the victims and claimed Wednesday's fatal blaze was due to negligence, with many citing the new cladding put on the 1974 concrete tower.
Speaking later outside Scotland Yard, Mr Cundy added: 'I have investigated major crime for most of my service and I have seen some bad things. I wonder now if they might have been better left in their beds.
Meanwhile, Queen Elizabeth II on Saturday led a two-minute silence in the memory of the victims who died in the tragic incident.
Cundy promised an "exhaustive" criminal investigation into the fire that would also look at a major refurbishment of the building completed previous year.