The former chancellor says he has a “moral responsibility” to help the poorest with energy bills.
Rishi Sunak has said he would rather lose the Tory leadership race than “win on a false promise”.
In a BBC interview, the former chancellor said he would tell people what “they needed to hear” and stay “true” to his values.
He said the next PM has a “moral responsibility” to support poorer households with targeted payments.
And he said a plan by his rival Liz Truss to help people though tax cuts would not help the most needy.
Speaking to the BBC’s Nick Robinson, he said he would spend billions of pounds on further targeted payments to pensioners and those on low incomes.
The payments would come on top of a £15bn package of payments he announced as chancellor in May.
Mr Sunak said it was too early to put a precise figure on the extra payments to help people struggling with their energy bills.
However, he suggested they would compensate for a projected £400 increase in bills compared to predictions earlier this year.
How to help households with rising energy bills has become a key dividing line in the contest to succeed Boris Johnson as next UK prime minister.
It comes after a forecast on Tuesday suggested bills for a typical household could hit £4,266 next year, higher than predictions made earlier this year.
Mr Sunak said he would “go further in supporting those families that most need our help” because the predictions had become worse.
“That’s why I do feel a moral responsibility as PM to go further, and get extra help to people over the autumn and the winter,” he added.
“I think that’s the right priority, that’s what my plan is about.
“Alternative proposals that don’t focus on that I don’t think are right, and I don’t think are actually the moral thing to do.”
The BBC has also invited Liz Truss to a one-on-one interview with Nick Robinson and says it has been discussing timings with her team.
Analysis by David Cornock, BBC political correspondent
Both Rishi Sunak and Liz Truss believe they are on the right side of the moral argument over the cost of living crisis that is dominating this leadership election.
Liz Truss argues that putting up taxes is immoral – but Rishi Sunak told Nick Robinson the government had a “moral responsibility” to go further to help those who cannot afford to heat their homes.
He suggested bills could be £400 higher than anticipated when he was chancellor. How much further would he go? Anyone hoping for specific figures will have to wait but there were strong hints of targeted help for pensioners and those on the lowest incomes.
This interview may be seen and heard by millions but both candidates have tried, to varying degrees, to target their message at the 160,000 or so Conservative Party members who will choose our next prime minister.
Mr Sunak, the self-styled underdog in this contest, appeared to acknowledge that those members may prefer to hear Liz Truss’s message on tax cuts.
He said he went into the contest prepared to tell people to tell people “what they needed to hear rather than what they needed to hear, not necessarily what they wanted to hear”.
That doesn’t mean he hasn’t refined his own message during the campaign – he’s promising to cut VAT on energy bills, a position he rejected in government – but he made clear he would rather lose (or win) on his own terms.
“Our Next Prime Minister: The Interviews” is available on the BBC Iplayer and is on BBC One at 7pm