In a new dire outlook for households, Cornwall Insight said bills are set to soar to around £3,582 in October, from £1,971 on Tuesday, before rising even further in the new year.
The Sun reported gas and electricity executives will meet with the Cabinet ministers on Thursday morning, when the bosses will be asked to submit a breakdown of expected profits and payouts as well as investment plans for the next three years.
At a leadership hustings in Darlington on Tuesday, Ms Truss said that she "fundamentally" disagrees with "putting up taxes and then also giving out benefits" to help with the rising cost of living.
Asked about what she would do to deal with rising fuel prices, the South West Norfolk MP told the hustings audience: "I understand people are struggling with their bills on fuel and food but the first thing we should do as Conservatives is help people have more of their own money.
"What I don't support is taking money off people in tax and then giving it back to them in handouts. That to me is Gordon Brown economics."
Asked if she would get in a room with the Prime Minister and her leadership rival to look at providing additional support, Ms Truss said her focus in Government is on her role as Foreign Secretary.
She said the Prime Minister and Chancellor Nadhim Zahawi are "capable people, capable of making these decisions", adding "this kangaroo committee you're proposing sounds bizarre".
The former chancellor also suggested he would not offer further cash payments to every household and would instead target support at the most vulnerable.
Mr Sunak added: "The only way to help them is with direct support because tax cuts alone are not much good if you're a pensioner who is not earning any extra money."
Asked if he is planning support similar to earlier plans of providing £400 to every household regardless of their income, he said: "No, because I think what we need to do is target our support for the most vulnerable."
The pair will face further scrutiny over their plans for the future of the UK on Wednesday, with Mr Sunak due to be interviewed on BBC One, while Ms Truss will take questions from an audience of GB News viewers in Greater Manchester.
Boris Johnson meanwhile declared he is "absolutely confident" that his successor as prime minister "will have the fiscal firepower and the headroom to continue to continue to look after people".
The Government has already promised £400 to every household and extra help for the more vulnerable.
On Monday, Scotland’s First Minister called on Mr Johnson to move a proposed meeting between the heads of the UK’s governments from September to this week to address the cost-of-living crisis.