Around 200 Metropolitan Police officers continue to work on the criminal inquiry into the fire, which killed 71 people last June.
Detectives are considering manslaughter, corporate manslaughter and misconduct charges, as well as breaches of fire safety regulations, as part of what has been called an investigation of "unprecedented" scale and complexity.
The force has requested the Home Office pay the £27m cost of the inquiry in the coming year, their finance chief said on Thursday.
A further £11.1m was also being sought to cover overtime investigators have already worked.
Current rules allow police forces to apply for grants to help pay for an investigation once it exceeds 1 per cent of their budget.
Given how many officers Scotland Yard says are required to participate in the second year of the investigation, it has asked the Government to provide more funding.
Lynda McMullan, the Met'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 per cent 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."
She added: "We don't think it would be reasonable for us to pick up the full costs, as we know that investigation will go on for some time."
Home Office officials have told the force the matter will be put to ministers, who will decide whether to approve the additional funding, the committee heard.
Suspects in the Met's investigation are not expected to be interviewed until later this year, while Scotland Yard expects to hand the tower – which is still a crime scene – back to Kensington and Chelsea Council in the spring.
A spokeswoman for the Home Office said it would consider the application "as a matter of priority".
Agencies contributed to this report
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