His team said the approach will aim to replicate previous measures used to fund support to Ukraine.
This resulted in departments and devolved administrations being asked to find missing expenditures from their capital budgets, which includes money spent on investment and things used to create future growth.
Over the coming months, and at a time when Truss had to get on the defensive, Sunak sought to refocus her focus on the cost of living amid renewed concerns about energy bills.
Last week, the Bank of England announced its forecasts that inflation could rise by more than 13% in October, while another forecast found that the energy price ceiling could rise to £3,358 annually from October and reach £3,615 from January.
If Truss wins the race for number 10 over the weekend, the secretary of state has suggested that he won’t be able to support families in distress and that his priority is to reduce the tax burden.
But his allies insisted he was committed to helping families struggling with rising bills.
Earlier, Liberal Democrats said the October energy bill price ceiling increase should be canceled by the next prime minister to help households save hundreds of pounds.
Party leader Ed Davey said the government must close the gap in energy suppliers to ensure they can supply customers at current rates.
The party estimated the cost of the policy to be £36 billion and suggested that the unexpected tax on the profits of oil and gas companies should be expanded to help meet this.
The two Tory leadership hopefuls will continue the campaign on Tuesday before a rush session with party members in Darlington, an area called the “red wall” under Boris Johnson’s 2019 leadership.
Sunak described Truss’s contingency budget tax cut plan as a “big problem” for big businesses and the better off, adding that he will do little to help those most in need over the next winter.
Regarding his plans, Sunak said: “I have no doubt that this winter will be extremely difficult for families across the country and more support will be needed.
“In the spring, I identified a set of measures to provide assistance by targeting the most vulnerable to the most support. But the bills are rising more than expected and the next government will need to act.
“As Chancellor, I created a framework to support hardworking families and retirees to lower the bills. People need proven methods that will deliver for them quickly. So I will use the framework I created to provide more support and give millions of people the peace of mind they desperately need before winter.
“I am very clear on what is needed to help people and will take action as soon as I know how much the bills will go up.
“It’s important that people know how to pay for this extra support. To keep one-off borrowing to an absolute minimum, I will first seek efficiency savings across Whitehall to provide direct support to families dealing with the unprecedented situation we face.”
A Truss campaign source said: “Rishi’s question about the economics of new donation commitments is: How will he fund these new pledges?
“Three weeks ago he was saying that borrowing more is irresponsible and inflationary. Has he changed his mind? It feels like another big U-turn.
“How can Liz’s tax cuts be if Rishi’s borrowing isn’t inflationary? Intellectually it is as waterproof as a sieve.”
Scotland’s first minister, Nicola Sturgeon, has called for an emergency meeting between the heads of the devolved administrations to address the cost of living crisis.
Meanwhile, Tony Danker, director general of the Confederation of British Industry, warned the government that it should have “all hands at the pump”, while former prime minister Gordon Brown called for quick response to address rising energy costs.
But Downing Street said Johnson had no plans to introduce major new fiscal measures before the end of his prime ministership.