But Sunak’s campaign said the move would “incredibly regressive” and cost more than £30bn.
Another option reportedly being weighed up by Truss is a cut to income tax, with proposals including increasing the level above which people start paying the levy.
Truss could also raise the tipping point for the higher rate of 40% and cut the basic rate below 20%.
Boris Johnson has said whoever succeeds him in No.10 would announce “another huge package of financial support” amid soaring energy bills.
The outgoing prime minister has refused to act again himself, deciding the country must wait until a new PM is in office to provide any new help.
The Treasury has said it is making the “necessary preparations” to ensure the next government has options to deliver extra help “as quickly as possible”.
Energy regulator Ofgem warned the government on Friday that it must act urgently to “match the scale of the crisis we have before us” as Britain faced the news that the average household’s yearly bill will rise from £1,971 to £3,549.
Sunak has already said he will provide additional support targeted at the most vulnerable.
Truss has promised “decisive action” to deliver “immediate support” if she wins the contest.
But she has so far been vague about what form this assistance might take apart from slashing green levies on energy bills and reversing the controversial National Insurance hike.
In an article for Mail+, Johnson acknowledged that the next few months will be difficult – “perhaps very tough” – as “eye-watering” energy bills take their toll, but he forecast the UK will emerge “stronger and more prosperous (on) the other side”.
He said “colossal sums of taxpayers’ money” have already been committed to assisting people with their bills.
But he added: “Next month – whoever takes over from me – the government will announce another huge package of financial support.”