Police will look at the policy as part of their criminal investigation into the blaze.
The announcement came just minutes after legal representatives for firefighters told the Grenfell Tower Inquiry that there was “no obvious and safe alternative strategy” to the stay put policy.
In the first two hours of the June 14 blaze, operators and fire personnel advised residents of the tower to remain in their flats - a policy that has since been the focus of intense scrutiny.
The police’s criminal investigation is taking place concurrently but separately to the public inquiry.
Asked if senior fire officers could be charged with manslaughter for not telling people to leave, Detective Superintendent Matt Bonner said on Thursday that a prosecution could most likely fall under health and safety legislation.
DS Bonner said: “The LFB would, as any other organisation involved, have an obligation to conduct their activity in a manner that doesn’t place people at risk.
“It doesn’t mean that at the moment they have or they haven’t, but that’s where the legislation is most likely to arise if that was an eventuality.”
Bonner said that investigators had identified 507 companies or organisations linked to Grenfell Tower, 36 of which they said were the most significant.
Police have interviewed 579 firefighters and control room staff, and have 250 more to speak to.
All the survivors, except for most of the 65 children in the tower on the night, have been interviewed, detectives said.
On Monday, an expert report authored by Dr Barbara Lane, which was submitted to the inquiry, found that all residents should have been evacuated from the building much earlier after the stay put strategy failed within about 30 minutes.
The first 999 call to the emergency services was made at 12.54am by Behailu Kebede, the occupant of the fourth floor flat where the fire began.
Danny Friedman QC, speaking on behalf of a group of law firms representing survivors and the bereaved, said on Monday that by 1.15am the strategy was considered to have begun to fail and by 1.26am it had “substantially failed.”
Dr Lane, a leading fire safety expert, said she was “particularly concerned” by the delay after 2am, when a major incident was declared, until 2.47am before the advice was officially changed to total evacuation.
During the final day of opening statements on Thursday, Stephen Walsh QC, representing the London Fire Brigade (LFB), defended the decision of fire personnel to relay the stay put advice to residents and said the design of Grenfell Tower meant that it was not designed to facilitate a mass evacuation.
“Fire safety is a crucial element of the design process which frequently dictates the way in which the fire services are expected to carry out fire and rescue operations.
“Buildings such as Grenfell Tower were expressly designed so as to contain any fire in its compartment of origin for sufficient time to allow the fire service to extinguish it before it spread,” Walsh told the inquiry.
“The building design is not intended to facilitate simultaneous evacuation of the whole building, especially at the same time as firefighting.”
The FBU said that it should be considered whether early incident commanders had training and procedures on issues such as tackling cladding fires, looking out for signs of a breach of compartmentation and when to abandon a stay put policy.
Martin Seaward, representing the union, said: “We submit on the evidence so far that there were no such procedures, nor had firefighters received any such training.”
Seaward urged the inquiry not to judge the strategies employed on the night purely with the benefit of hindsight - a view echoed by Walsh in his opening statement.
Rounding off this week’s opening statements, counsel to the inquiry Richard Millett QC said he believes it “may very well be possible” for “urgent, interim recommendations” to be made ahead of the publication of inquiry chairman Sir Martin Moore-Bick’s first report.
Millett said that there was a “broad measure of agreement” among core participants for the move following calls from several legal representatives this week that important safety recommendations be made as soon as possible.
Hearings will resume on Monday, June 18.