When asked about further support for the cost of living, the former chancellor said: “That’s what my plan is about and actually, alternative proposals that don’t focus on that I don’t think are right and I don’t think are, actually, quite frankly the moral thing to do.”
Mr Sunak said extra support he would give to vulnerable households could total a few billion pounds.
“Right, because that’s the scale, if you look at the scale of the problem we’re talking about, and then it depends exactly how we spread that around and provide most help to those who most need it,” he said.
Mr Sunak said the government had a “moral responsibility” to help families through the crisis.
When he was chancellor in May he announced a £15.8 billion ($19.3bn) support package but on Wednesday he said that if he became prime minister he would go further.
“I do feel a moral responsibility as prime minister to go further and get extra help to people over the autumn and the winter, to help them cope with what is going to be a really difficult time,” Mr Sunak said.
“I think that is the right priority.”
Asked whether the extra payments would cost a “few billion” or more than £10 billion, he replied: “It’s much closer to the former than the latter.”
Mr Sunak also said he would “rather lose” the Conservative leadership contest while maintaining his values than promise “false things that I can’t deliver”.
He said he wanted to win back trust in politics after departing prime minister Boris Johnson’s turbulent leadership.
“I do think there has been a breakdown in trust because of the issues of the last several months and part of why I am sitting here is because I want to restore trust back into politics," Mr Sunak said.
“I want to put integrity and honesty at the heart of how I run government and how I want to be prime minister. And as you can see in this leadership contest, I have been doing that.
“I haven’t been saying the easy things and actually I am prepared to lose this contest if it means that I have been true to my values and I am fighting for the things that I think are right for this country.
“I would rather lose on those terms than win by promising false things that I can’t deliver.”
UK Conservatives on the leadership campaign trail — in pictures
But the Truss campaign accused Mr Sunak of “old fashioned, Gordon Brown-style politics of envy”.
“Liz believes in people keeping more of their own money, not Rishi’s socialist tax and spend, which will lead us to recession,” a campaign spokesman said.
Earlier, Treasury Chief Secretary Simon Clarke confirmed that officials were working on a new support package for the next prime minister to consider.
Mr Clarke said it was “absolutely right” for the new leader to consider “these options in the round” when they took the reins, and that they should steer clear of announcing “new, uncosted policies” during the election period.
Everything you need to know about Rishi Sunak - video
Meanwhile, Ms Truss — who had opposed further “handouts” — denied that she had ruled out more direct help for families struggling with soaring energy bills.
She said that if she became prime minister on September 5, she would instruct the chancellor to look at the issue “in the round” and to come up with a solution.
At the same time, Ms Truss said it was wrong to “whack up” taxes simply to pay out more in benefits, while her campaign team accused Mr Sunak of pursuing “socialist tax and spend” policies.
With analysts predicting average annual household energy bills could top £4,200 in January, the cost of living continued to dominate the exchanges in the battle to succeed Mr Johnson.
Ms Truss, who has said her priority is to reduce taxes, pledging to reverse a rise in national insurance, insisted she was committed to supporting working families through the winter.
“If the only answer to everything is to whack up taxes and give out more benefits, then the country is going to run into trouble," she said, taking questions on GB News from an audience in Leigh, Greater Manchester.
“We are predicted to have a recession. It is important we grow the economy.
“I will do everything I can to support working families who work hard and do the right thing.
“If I am elected as your prime minister, I will make sure the chancellor has an emergency budget and looks at this issue in the round and sorts this issue out.”
Everything you need to know about Liz Truss - video
Mr Sunak’s supporters said that she appeared to be backing away from her previous position.
“This is a major U-turn on the biggest issue currently facing the country,” a campaign spokesman said.
“Taking action means providing direct support, which Truss had previously dismissed as ‘handouts’.”