Liz Truss has pledged to cut taxes “immediately” if she wins the race for No 10, a plan condemned by rival Rishi Sunak as insufficient to help the most vulnerable amid the worsening cost-of-living crisis.

Ms Truss said she would use a September emergency budget to reverse the national insurance rate rise brought in by Mr Sunak when he was chancellor.

She would seek to implement the change within days, rather than wait until April in line with usual Treasury rules, it is understood.

The two candidates vying to replace Boris Johnson as prime minister continue to clash over their plans to help families with spiralling bills after the Bank of England warned that the UK would fall into the longest recession since the financial crisis, with inflation set to soar to more than 13%.

There are growing calls for the new premier to urgently increase the amount of support available to low-income households, with a new report commissioned by former prime minister Gordon Brown suggesting Government help has failed to address their needs.

Ms Truss wrote in the Sunday Telegraph: “I would hit the ground running by bringing in an emergency budget, charting a firm course to get our economy growing in order to help fund our public services and NHS.

Her tax proposals are not going to help very significantly people like pensioners or those on low incomes

“I would use this 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.”

The Foreign Secretary earlier insisted tax cuts, not “handouts”, would help families with rocketing fuel bills this winter.

Mr Sunak said: “It’s simply wrong to rule out further direct support at this time as Liz Truss has done, and what’s more, her tax proposals are not going to help very significantly people like pensioners or those on low incomes who are exactly the kind of families that are going to need help.”

Trade minister Penny Mordaunt denied that Ms Truss had ruled out expanding direct payments to people.

Ms Mordaunt, a former Tory leadership candidate who has thrown her weight behind the Foreign Secretary, told Sky News: “It’s not that she’s ruling out all future help, that’s a misinterpretation of what she said.

“What she is looking at though is enabling people to keep more of the money that they earn.

“It makes no sense to take money off of people and then to give it back in very, very complicated ways.

“We need to simplify this and we need to ensure that households are as resilient as possible and stopping taking large sums of tax off people is one way of doing that.”

Ms Mordaunt said tax cuts “don’t have to be inflationary”, in response to Mr Sunak’s argument that they would further drive up prices.

In an interview with The Times, Mr Sunak said: “The priority for me is to not do things that make it worse and I think putting £40 billion plus and borrowed money into an economy that’s seeing an inflation spiral does risk making it worse.

“It might be OK but I think it means taking a gamble with people’s savings, their pensions, their mortgage rates, it’s not a gamble I’m prepared to take so I don’t want to make it worse.”

The former chancellor told the newspaper the public deserved “clear-eyed realism and not starry-eyed boosterism”.

Allies of both camps criticised the other side’s plans, with former Tory party co-chairman Oliver Dowden labelling Ms Truss’s proposed tax cuts “insufficient”.

The Sunak supporter told BBC News: “You’re going to see energy bills going up to almost £4,000 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.

“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.”

Senior Conservative MP Mel Stride tweeted: “Pensioners typically won’t benefit. Workers on Living Wage gain under £60 a year. Millions of poorest will get nothing. Rishi’s right we need clear targeted support. Get this wrong and we lose next election.”

But former cabinet minister John Redwood, a backer of Ms Truss, tweeted: “Rishi now wants the state to spend and borrow more to help with the cost of living. Mr Flip Flop will discover the more you change your message the fewer people will believe anything you say. He told us if the state borrowed more we would get more inflation so he wouldn’t do that.”

Mr Brown pressed both candidates to agree an emergency budget with Mr Johnson this week or risk “condemning millions of vulnerable and blameless children and pensioners to a winter of dire poverty”.

“The reality is grim and undeniable: a financial timebomb 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,” the former prime minister wrote in Sunday’s Observer.

His report found existing Government support for low-income households has fallen short of offsetting the losses they face, with some families up to £1,600 worse off a year.

The additional £1,200 offered to the poorest in society this year will fail to compensate for three major blows to their income from October 2021 to October 2022, the analysis suggests.

More than half of the public think the Government could do more to help with rising costs but is choosing not to do so, according to a poll by opinion research agency Public First.

The survey of 2,011 people also found that 64% are finding it difficult or impossible to pay energy bills.

Tory MP Damian Hinds conceded the package of support Mr Sunak drew up as chancellor was not enough in these “extraordinarily difficult times”, and suggested more would come if he became prime minister.

He told Sky News: “Things have been getting worse even since that was put into place in terms of projections for energy bills… and he’s been clear that more may well be needed and and he is ready to do that as required.”