Energy Secretary Ed Davey has urged the energy regulator to look into the profits being made by the Big Six energy companies through supplying gas.
This comes as the Commons Energy and Climate change committee urged the minister to slap power companies with fines for their "complacency" during the Christmas black-outs and for not restoring power quickly enough.
Davey called on Ofgem to examine whether the energy giants' profit margins - in some cases five times higher for gas than for supplying household electricity - should be the subject of a market investigation.
Such an investigation, if it found evidence of a monopoly, could see a company being broken up.
He claimed gas made up two-thirds of the energy bills households connected to the grid and that if profit margins for gas came down to a similar level to those in the electricity market it could save every household £40 per year.
The recommendation came in a letter to Ofgem chief executive Andrew Wright, in which Davey singled out British Gas.
He said there was evidence the company, which has the greatest share of the domestic gas market, had tended "to charge one of the highest prices over the past three years, and has been on average the most profitable".
Davey wrote: "Clearly you will wish to consider whether this is prima facie evidence of an issue in the market and so whether it merits a market investigation reference with the whole gamut of potential remedies that could follow including a break up of any companies found to have monopoly power to the detriment of the consumer.
"Alternatively you may of course conclude that no action is needed or potentially some intermediate measure which can be taken by the sector regulator."
Davey's letter also suggests looking at whether the Big Six should adopt a different business model and how well British energy markets are linked to those in mainland Europe.
Which? Executive director, Richard Lloyd, said: "This is a hugely significant intervention by the Secretary of State for Energy and implies that Ed Davey agrees with Which? that the structure of the biggest energy companies is partly to blame for the price hikes that millions of squeezed customers have been struggling with in recent years.
"The pressure is now on the regulators to announce next month that they are taking the first steps towards a more radical reform of the energy market, and to give consumers confidence that they are paying a fair price."
British Gas said it was complying with an ongoing independent market assessment being conducted by the Office of Fair Trading (OFT), Ofgem and the Competition & Markets Authority (CMA).
"Further discussions have been arranged over the coming weeks in which we will fully participate," a spokeswoman said.
"The data referred to in the Secretary of State's letter has already been fully disclosed and in the public domain for a number of months.
Shadow energy and climate change minister Jonathan Reynolds accused Davey of spending months defending energy firms despite evidence of overcharging.
"Actions speak louder than words - and this Government has let the energy companies get away with increasing their profits on the back of spiralling bills for hard-pressed consumers," he said.
"If the Government wants to be taken seriously on energy bills, nothing less than a price freeze and action to stop these firms from overcharging in the future, as Labour has proposed, will do."
Ofgem said it would look at all evidence while compiling its report with the CMA and OFT. It declined to comment further.