Prime Minister Andy Burnham has asked ministers to develop a plan to remove standing charges from gas bills.
These are fixed daily fees that cover the costs of connecting to a supply. They vary by region and households must still pay them even when energy is not being used.
Industry body Energy UK said whether the proposed changes would make a difference to bills would depend on how or where the charges were reallocated on the bill.
But an influential research foundation argued it would be a progressive move that would help those on low-incomes and knock £27 off a typical annual bill.
During Prime Minister's Questions, Burnham said the "pragmatic decision" on giving the green light to the Jackdaw gas field in the North Sea would allow the government to increase investment in renewables and "do more to help people with bills".
He said he had asked the Chancellor John Healey and Energy Secretary Miatta Fahnbulleh "to look at whether we can remove standing charges from gas bills and to report back to this house at the earliest opportunity".
No details have been given on how removing standing charges would work in practice. But the suggestion has been made by research firm Nesta that they could be moved to the unit rate, external, so people pay for what they use.
This would probably benefit customers who don't use much gas, but heavy users could end up paying more.
For nearly two years, regulator Ofgem has run a series of consultations on standing charges, with a massive public response including vociferous opposition from low-energy users who can do nothing to cut standing charges.
Ofgem is running a pilot scheme on low standing charge tariffs, but is yet to draw any firm conclusions, suggesting there are no easy answers.
Average standing charges are 54.8p a day for electricity and 29.7p a day for gas.
When asked why electricity standing charges would not be scrapped as well, the PM's official spokesman said VAT had already been cut on electricity bills.
Standing charges cover costs such as transporting energy to homes via pipes and wires, and the upkeep of that infrastructure.
The standing charges also fund schemes to support low-income households.
Many of these costs will need to be recouped somehow, but Citizens Advice has warned, external about the potential consequences of reform for vulnerable customers.
Vulnerable people who don't have a high income, but have high energy usage, could be penalised.
But Madeleine Gabriel, director of sustainable future at Nesta, said: "Abolishing the gas standing charge and moving those costs onto the unit rate would help those worst hit by rising energy prices."
Pressure remains on the chancellor to offer more fundamental, targeted energy support this winter in his Budget next week, with energy prices forecast to rise by 16% to a four-year high in January.
In addition to standing charges, Burnham said investing in renewables would also bring household bills down "substantially" by breaking the link between gas and electricity prices "in this parliament".
Electricity prices in the UK are mainly set by gas prices, and this is often cited as a reason why electricity prices in the UK are higher than many other countries.