A leaked video showed Mr Sunak boasting Conservative Party members in wealthy Tunbridge Wells that he had instead diverted money to towns like theirs.
Labor has now written to the equalization secretary urging him to investigate the comments.
In her letter to Greg Clark, his Labor shadow Lisa Nandy said: “This is deeply worrying. The former chancellor admits to setting the rules to channel taxpayers’ money to wealthy parts of the country at the expense of areas he himself admits are ‘deprived’ .
“Public money must always be distributed fairly and used in areas where it is most needed. It is completely unacceptable that ministers allocate taxpayers’ funds to ensure political gain for themselves or their party.”
She urged him to investigate as soon as possible the rationale for the changes and what assessment was made of their impact.
She also demanded that his department publish measures that had been put in place to ensure that the allocation of funds from the second round of the Leveling Up Fund was fair.
Ministers have been accused of using it and other funds to prop up seats the party wanted to win at the next general election.
In the recordings, obtained by The new statesmanMr Sunak claimed that Treasury formulas “pushed all the funding into deprived urban areas” in Britain before he became chancellor.
He said he had started to change how the money was allocated to ensure more places like the south-east city received “the funding they deserve”.
Senior Tories also criticized Mr Sunak over the comments.
Foreign Secretary Zac Goldsmith said: “This is one of the strangest – and stupidest – things I’ve ever heard from a politician.”
Jake Berry, the chairman of the Northern Research Group of Tory MPs, said Sunak publicly “claims he wants to level up the north – but here he is bragging about trying to divert vital investment away from deprived areas”.
But Sunak’s supporters defended his remarks. Conservative Tees Valley mayor Ben Houchen claimed Boris Johnson had led the party to victory in 2019 on a promise to invest in areas “that have been ignored at the expense of cities”.
The row comes just a day after it emerged Mr Sunak’s rival for the keys to No 10, Liz Truss, faced the possibility of her first major sleaze inquiry amid claims she failed to declare “dark donations” to his leadership campaign.
Labor has asked the Cabinet Secretary to open an inquiry into the funding of a so-called “Fizz with Liz” champagne dinner.