The idea of business-friendly zones with low taxes and regulation is favoured by many Conservative MPs as a way to boost growth and make the best of Brexit.
Mr Sunak talks up the fact that he announced the creation of the first eight last year, when he was chancellor, and has promised to bring more to Wales and Northern Ireland if he wins the Tory race.
In a recent hustings, he went as far as to say that "I came up with the idea of free ports", which raised eyebrows because the Conservatives called for them as long ago as the 1983 election when Mr Sunak was three years old.
Ms Truss, meanwhile, sought to outflank him by promising new investment zones, or "full-fat free ports" as she calls them. She said these would come with new "model towns" such as the Victorian-era chocolate maker's home of Bournville.
Visiting the Commonwealth Games in Birmingham on Saturday, Ms Truss promised investments in regional transport and heralded her support among prominent Conservatives in areas that might stand to benefit.
"Liz Truss understands that our regions represent a potential gold mine of growth, the West Midlands especially," said Andy Street, the Tory mayor of that region. "Her plan for investment zones, for instance, will give us a powerful tool for unlocking new investment, jobs, and growth."
The Sunak campaign hit back by saying Ms Truss was borrowing policies already introduced by the government or promised by Mr Sunak if he wins the contest.
"Not only are Team Truss copy and pasting policies put in place by Rishi himself, but they are also re-announcing two-year-old government policies," a spokesman for Mr Sunak said.
"Imitation is the best form of flattery, as the saying goes."
Another piece of evidence called in aid by the Sunak campaign is that their man wrote a paper in 2016 called The Free Ports Opportunity, setting out one of the possible benefits of Brexit.
But this history did not go unchallenged. Daniel Korski, a former Downing Street aide, unearthed a photo showing that both candidates had co-chaired the first meeting of a free port advisory panel in 2019. Ms Truss was trade secretary and Mr Sunak a junior treasury minister at the time.
"[The] truth is they were a great double-act," said Mr Korski, who also ran Tom Tugendhat's leadership campaign earlier in the contest. However, Mr Korski said it was fair to say that Mr Sunak had "reinvigorated discussions" about free ports.
Mr Sunak's team also questioned the comparison to Bournville. They posed the somewhat tongue-in-cheek question of whether this would involve companies owning their workers' homes as the Cadbury company did for its employees.
But a Ms Truss supporter, northern MP Jake Berry, said the foreign secretary was the candidate to deliver on flagship Conservative promises to "level up" Britain's north-south divide.
"Politicians in Westminster have often sung a good song about addressing regional inequality, with varying degrees of success," Mr Berry said. "I firmly believe that Liz’s plan for levelling up will deliver a bright future for the north, and the entire United Kingdom."
Ms Truss opposed Brexit at the time of the referendum in 2016, but now says she has converted to the cause and has promised to slash regulations inherited from the EU.
Both candidates are promising plans for growth as Britain faces increasing economic woes amid runaway inflation and predictions of a looming recession.
The two candidates clashed in Eastbourne, on the south coast of England, on Friday in the fourth out of 12 official hustings before a new Conservative leader is announced on September 5.
That person will replace Boris Johnson as prime minister the following day.
Conservative leadership race - in pictures