The winner of the leadership contest will be announced at 12:30 BST before they take up office on Tuesday.
The new PM inherits a flagging economy, with inflation at a 40-year high.
BBC News has been told that a menu of options has been worked up in Whitehall to help struggling households, including a freeze on energy bills.
Industry sources are increasingly optimistic that the new prime minister will back plans to freeze the energy cap, the maximum price for domestic gas and electricity set every three months by regulator Ofgem.
This would not necessarily require upfront government financing at the beginning, BBC News economics editor Faisal Islam reported.
It follows multiple meetings with government, including ministers close to Foreign Secretary Liz Truss.
Ms Truss, tipped by pollsters to win the contest, has promised to announce further help to shield consumers within a week of taking over.
She then plans to deliver £30bn in tax cuts through an emergency Budget later this month, arguing the UK's tax burden is behind sluggish growth.
Her rival, former chancellor Mr Sunak, has signalled he believes he has lost, saying his job "now is just to support a Conservative government".
Ms Truss is yet to offer details of her cost-of-living support plan beyond saying she will temporarily scrap green levies on energy bills and reverse the rise in National Insurance introduced during Mr Johnson's tenure.
Mr Sunak announced payments of £15bn as chancellor, including £400 payments for all households, but both contenders have said further support will be required after cost predictions rose further over the summer.
On Sunday the foreign secretary declined to say whether further help would be universal or targeted at the most needy, saying she would need time in office to iron out the details of her plan.
She admitted unpicking the National Insurance rise would benefit higher earners more, but said it was justified because it would boost the economy overall.
Speaking to the BBC's Laura Kuenssberg, she blamed a focus on distributing wealth through taxes for low economic growth over the past two decades.
Labour, the Liberal Democrats and the SNP have all called on the government to freeze energy prices through multi-billion pound subsidies, while the Greens have suggested nationalising the UK's five biggest suppliers.
Ms Truss did not rule out a freeze on Sunday but has previously described the idea as a "sticking plaster" and argues more needs to be done to help the UK boost its domestic sources of energy.
She said any further support would have to go "hand in hand" with efforts to boost nuclear energy, fracking for shale gas and more oil and gas drilling in the North Sea.
As well as tax cuts, Ms Truss has pledged to deliver low-regulation investment zones and bring about the biggest increase in defence spending in decades.
She abandoned a plan to link public sector pay to local living costs, however, after a backlash from unions, Labour and some Tories.
The seven-week leadership contest brings to an end Mr Johnson's turbulent three years in office, and has seen the candidates regularly attack each other's policies as well as the Tories' record in government.
Mr Johnson was forced out in July by a ministerial revolt over a string of scandals, just over two-and-a-half years after leading the Tories to a landslide victory at the 2019 election.
The original field of 11 contenders was whittled down to two in a series of Tory MP ballots, with the final pair going into a run-off to be decided by the membership, which stands at about 160,000.
Although Mr Sunak had the most support among Tory MPs, he has trailed Ms Truss in opinion polls of the party grassroots.
Mr Johnson is expected to deliver a farewell speech upon leaving office on Tuesday, before the handover of power takes place.
In a break with tradition, the next Tory leader will travel to Balmoral Castle in Scotland to be appointed by the Queen, rather than at Buckingham Palace.
The Queen has been suffering from mobility issues and it is understood the change announced last week was made to prevent the need for any last-minute rearrangements.
-- Courtesy of BBC News