The London force has submitted an £11.1m claim to cover for overtime and extra costs in 2017-18.
They will also request a further £27m to fund the 'entire' costs of the investigation, which involves 200 officers and staff, for 12 months as of April 2018.
Around 200 officers from the Metropolitan Police are continuing working on the criminal probe into the fire, which killed 71 last June.
Lynda McMullan, the Met Police’s director of finance, said: "We don't think it's reasonable for us to pick up all the costs."
Under current rules, a police force can apply for a special grant to help foot the bill of an investigation once it exceeds 1% of their budget.
The size of the Grenfell Tower probe has previously been described as “unprecedented”
Last month, a public inquiry into the Grenfell Tower fire, headed by Sir Martin Moore-Bick, heard that the Met was investigating offences including manslaughter, corporate manslaughter, misconduct in public office and breaches of fire safety regulations in relation to the fire.
The force has gathered 31 million documents and 2,500 physical exhibits.
Some 1,144 witnesses have given statements and 383 companies are part of the investigation.
The Met has also requested extra government funding of £7.8m to help pay for policing following last year's terror attacks.
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