Boris Johnson has acknowledged that the package of support to help people pay surging energy bills is not enough, as the split deepens between Rishi Sunak and Liz Truss over how to deal with the crisis.

Amid forecasts that energy bills could hit £5,000 a year by next April, the prime minister made clear that he expected his successor to act.

Asked if the current package of £400 off bills, rising to £650 for vulnerable households, was enough, Johnson said: “No, because what I’m saying what we’re doing in addition is trying to make sure that by October, by January, there is further support, and what the government will be doing, whoever is the prime minister, is making sure there is extra cash to help people.”

Johnson said earlier this week that he was sure his successor would offer more help with the cost of living, but Truss has held off spelling out how she would act and repeatedly dismissed the idea of “Gordon Brown style” “handouts”. Her stance appears to have slightly softened, with her campaign suggesting help would be forthcoming.

However, Thérèse Coffey, the work and pensions secretary and one of Truss’s key allies, said on Friday that the cost of bills would be “nowhere near” the levels predicted by independent analysts.

Defending Truss, Coffey said: “She is absolutely an MP who knows what it is like for struggling households and that is why, quite rightly in a considered way, once Ofgem comes up with their price cap … all of government … and it will be a decision for the new prime minister to enact what changes could be made.”

Sunak announced a £10bn package on Friday designed to help with energy bills this autumn, saying he could be open to “some limited and temporary one-off borrowing as a last resort to get us through this winter”.

“People need reassurance now about what we will do and I make no apology for concentrating on what matters most,” Sunak wrote in the Times, which reported that he valued a cut to VAT on energy at £5bn.

He was also said to have promised to find the same amount again to go towards helping those most in need, as he said: “You can’t heat your home with hope.”

The former chancellor also highlighted Truss’s refusal to say whether she would maintain the windfall tax on oil and gas profits, which is expected to bring in more than £5bn from oil and gas giants making money out of high prices.

In an interview with Times Radio, Sunak defended the windfall tax he had implemented as chancellor, saying it would “automatically raise more money” in tax to support struggling households as energy profits increase.

The former chancellor added: “I think that is the right thing to do and I think Liz Truss last night said she opposed doing that, and actually didn’t believe in that policy, so I think that is a question for her to answer.”

He also said his opponent’s plans to cut taxes would do “virtually nothing” for pensioners or the least well off.

A Sunak campaign spokesperson added that Truss had “blown a further £5bn black hole in her plans” by not backing the windfall tax.

Sunak also rejected the idea that Truss was the frontrunner and likely to win, saying that “lots of people have not made up their mind”.

After Thursday’s hustings, Truss reiterated her belief that tax cuts should be the main response to soaring bills. Truss told an audience of Tory members in Cheltenham that this would always be her “first port of call”, followed by a focus on longer-term energy supply issues such as support for fracking and nuclear power.

Truss said she could provide other assistance, but gave no details, saying she “can’t write the chancellor’s budget” before even being elected as prime minister.

“If the answer to every question is raising tax, we will choke off economic growth, and we will send ourselves to penury, and I think that’s a massive problem,” she said.