Their responses to the worsening economy have emerged as the main battleground in the bid to lead the country.
Ms Truss will take her campaign to Essex on Monday while Mr Sunak plans to court communities in the north-west of England. The pair will go head-to-head at two hustings this week — in Darlington on Tuesday and in Cheltenham on Thursday.
Ms Truss came under fire from Mr Sunak’s allies for suggesting there would be no “handouts” if she led the government, but later played down the comment.
The row followed the Conservative party leadership favourite telling the Financial Times she would “look at what more can be done” after warnings from the Bank of England about the longest recession since the financial crisis and inflation rising to more than 13 per cent.
“The way I would do things is in a Conservative way of lowering the tax burden, not giving out handouts,” Ms Truss said.
Mr Sunak quickly condemned it as “simply wrong to rule out further direct support” for struggling families this winter.
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Penny Mordaunt, a former Tory leadership contender who has thrown her support behind Ms Truss, insisted she had been misinterpreted.
“It’s not that she’s ruling out all future help — that’s a misinterpretation of what she said," Ms Mordaunt told Sky News.
“What she is looking at, though, is enabling people to keep more of the money that they earn."
But supporters of Mr Sunak accused Ms Truss of another about-face after she last week retreated on proposals to cut public sector pay outside London.
Former chief whip Mark Harper tweeted: “Stop blaming journalists [again] — reporting what you actually say isn’t ‘misinterpreted’.
“Second time in just five days. This kind of thing happened under the current PM and hugely damaged trust in us all.
“So just what does ‘not giving out handouts’ mean then?”
Ms Truss has also faced criticism of her plans to use a September emergency budget to immediately reverse the national insurance rate rise brought in by Mr Sunak when he was chancellor.
“I would use it to immediately tackle the cost-of-living crisis by cutting taxes, reversing the rise on national insurance and suspending the green levy on energy bills,” she wrote in The Sunday Telegraph.
Former Tory party co-chairman Oliver Dowden, who is backing Mr Sunak, said Ms Truss's proposed tax cuts were “insufficient” to help low-income workers and would disproportionately benefit those better off.
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“You’re going to see energy bills going up to almost £4,000 ($4,820) and if you look at the idea of the tax cuts, this idea of reversing the national insurance contributions, that’s only going to benefit someone working full-time on the national living wage by less than £60," Mr Dowden told the BBC.
“Contrast that with whoever the prime minister is, they’re going to get a benefit of about £1,800.
“So this isn’t the way to help people through this very difficult period.”
Mr Sunak would go further than the extra £1,200 he offered to the poorest in society as chancellor, his allies suggested. He has also pledged to axe VAT on fuel bills.
Mr Dowden told Sky News that Mr Truss “has put her emphasis on this tax cut”, while Mr Sunak accepts the need for “bold, big interventions” to ease the burden on cash-strapped families.
He said the crisis called for an “intervention on a considerable scale” and pointed out that changes could be made without an emergency budget.
“So if you look at the £1,200 that was announced earlier ... that wasn’t through an emergency budget, but I think there is no doubt that we do need an intervention of a considerable scale to deal with this, because we have to be honest with people about the scale of the challenge that they are facing," Mr Dowden said.
On the energy price cap, he added: “Let’s see when we know the exact level of the price cap, but if it looks like it’s going to be at that kind of scale then of course we’ll need further intervention.”
The price cap is expected to rise to £3,359 in October and to £3,616 in January 2023, according to forecasts by energy regulator Ofgem and analysts from Cornwall Insight.
Ofgem has said that households face a “very challenging winter” amid surging inflation caused by geopolitical tensions and uncertainty over Russian gas supplies.
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Tory MP Damian Hinds conceded the existing package was not enough in these “extraordinarily difficult times”.
“Things have been getting worse even since that was put into place in terms of projections for energy bills … and [Mr Sunak's] been clear that more may well be needed and he is ready to do that as required,” Mr Hinds told Sky News.
Writing in The Sun, Mr Sunak ripped apart his rival’s plan to deal with the cost-of-living crisis, saying it would “give a big bung to large businesses and the well-off, leaving those who most need help out in the cold”.
He claimed Ms Truss's plan to slash corporation tax would not benefit small businesses, but rather “put money back in the coffers of the biggest companies with the largest profits”.
“These tax cuts simply won’t touch the sides,” Mr Sunak said.
Gordon Brown, a former Labour prime minister, has called for an emergency budget before a “financial time bomb” explodes this winter.
He said that millions of people risk being pushed “over the edge” if the government does not address the crisis gripping the nation.
Mr Brown said Ms Truss and Mr Sunak should work with Boris Johnson this week to immediately agree a package, or Parliament should be recalled.
Writing in The Observer, Mr Brown said: "The reality is grim and undeniable: a financial time bomb will explode for families in October as a second round of fuel price rises in six months sends shock waves through every household and pushes millions over the edge."
UK Health Secretary Steve Barclay has said the National Health Service is braced for “very serious challenges” this winter.
As the health service continues to tackle huge backlogs more than two years after the Covid-19 pandemic struck, it is facing a triple threat: a potential new Covid wave, the winter flu season and the fallout from the cost-of-living crisis.
Ms Truss and Mr Sunak continue to criss-cross the country seeking support from Tory members to be elected as the next party leader and prime minister.
Voting has begun, with the result to be announced on September 5.