The force says the demands on police time could leave it facing unreasonable costs because of the complexity of the investigation.
About 200 Metropolitan Police officers have been working on the criminal probe into the fire which killed 71 people in June 2017.
Millions of documents have been seized and months have been spent removing potential evidence from the ruined tower block in west London, which remains a crime scene.
Many suspects have been interviewed, and detectives have been investigating potential offences including misconduct in public office, manslaughter, corporate manslaughter and breaches of fire safety regulations.
The force is asking the Home Office to fully fund the £27m cost of the inquiry in the coming financial year, its head of finance has said.
Another £11.1m is also being sought to cover the extra hours put in by detectives so far.
A police force can apply for a special grant to help foot the bill of an investigation once it exceeds 1% of their budget under current rules.
Image: Scotland Yard says 200 officers have been investigating Grenfell
But because the Met‘s investigation into the fire is likely to involve many officers for a second year, it has asked the Home Office for more cash.
The London Assembly‘s budget and performance committee was told the department had provided assurances their request was being given “serious consideration”.
Lynda McMullan, the Met Police‘s director of finance, told the hearing: “The impact of Grenfell, that has had a very significant impact and we‘ve put in a special grant claim for the current year for £11.1m, in terms of the additionality of cost to do with the incident for the current financial year.
“We have also indicated that we would like – we estimate that we will be spending – close to £27m on that particular investigation.
“We are asking whether or not we can have dispensation, not just simply the 1% referenced, but we also want to claim the full cost of our officer time for that investigation, not just the additionality, the overtime, in effect.
“We don‘t think it would be reasonable for us to pick up the full costs, as we know that the investigation will go on for some time.”
Video: 27 December: Grenfell operator hands homes back to council
The force has been told by Home Office officials that the matter will be put before ministers to decide whether to sign off on the additional funding, the committee heard.
Ms McMullan added: “We would like to have full costs rather than incremental costs for the next year because of the fact given, after a reduction in the number of officers, it doesn‘t seem reasonable we would absorb that full cost of those 200 officers.
“We have got assurances it is being given serious consideration.”
A Home Office spokeswoman said: “Given the unique and tragic circumstances of this event, we will consider any application we receive for funding from the Metropolitan Police as a matter of priority.”