The ruling Conservative Party is choosing a new leader after Johnson was forced to quit when dozens of ministers resigned in protest at a series of scandals and missteps. Party members are voting by post to select either Sunak or foreign minister Liz Truss.
Polling shows Sunak, who was finance minister between February 2020 and July 2022, is trailing Truss as the two candidates tour the country in a bid to secure votes.
His comments came in a video published on Friday by the New Statesman magazine, which it said was filmed on July 29 at a meeting of Conservative Party members in Tunbridge Wells, a relatively affluent area in south east England.
Sunak is seen telling an audience: "I managed to start changing the funding formulas to make sure that areas like this are getting the funding that they deserve, because we inherited a bunch of formulas from the Labour Party that shoved all the funding into deprived urban areas ... that needed to be undone. I started the work of undoing that."
The New Statesman did not specify who filmed the short video and Reuters could not independently verify the date or the location when the video was made. The context of his comments was not shown.
Asked about the comments, a source in Sunak's campaign referred to reforms to ensure rural areas received funding alongside urban centres, pointing to his efforts, announced in March 2020, to redraw rules the finance ministry used to allocate investment.
"Levelling up isn't just about city centres, it's also about towns and rural areas all over the country that need help too," the source said, of Johnson's policy to reduce regional inequalities.
"Travelling around the country, he's seen non-metropolitan areas that need better bus services, faster broadband, or high quality schools. That's what he'll deliver as Prime Minister."
Truss did not immediately comment on the video.
However, the opposition Labour Party seized on the video as evidence that the Conservatives, who have been in power since 2010, are not committed to spreading wealth across the country.
"Public money should always be distributed fairly and spent in areas where it is most needed," Labour's spokeswoman on Levelling up, Lisa Nandy, said in a letter to the government, describing the comments as "deeply concerning" and calling for an investigation.