Consumer Focus is calling for action from the industry regulator
A debate is developing after a consumer watchdog called for domestic energy bills to fall in tandem with plummeting oil prices.
A barrel of oil has fallen from a high of around $147 in July to around $70 and Consumer Focus wants to see domestic bills dropping as a result.
But the Energy Retail Association says bills will not fall immediately as gas wholesale prices remain high.
Petrol prices and airlines' fuel surcharges have been cut recently.
Fears of the global economy falling into recession have pushed down the price of crude oil.
This has led to the price of petrol for motorists dipping below £1 a litre at some retailers for the first time since December last year. British Airways and Virgin have also cut the fuel surcharge levied from some passengers.
But domestic energy bills, which are linked to the price of oil, show no sign of falling.
"Oil prices have been falling since July, yet consumers have seen unprecedented rises in their gas and electricity costs. Consumers must now be wondering why they are left waiting," said Ed Mayo, chief executive of watchdog Consumer Focus - a new merger of Energywatch and the National Consumer Council.
"We want immediate action from energy companies to reduce their prices in line with falling oil prices. This will be good not just for consumers, but for the whole economy."
He said regulator Ofgem should step in.
Consumer Focus said that gas prices have risen by 51% since the start of the year, and electricity bills up by 28% - putting the average annual household energy bill at £1,308.
However, energy analyst John Hall said that although the two were linked, gas prices did not rocket as much as oil prices earlier in the year and so now did not have so far to fall.
And Garry Felgate, of the Energy Retail Association, told the BBC's Working Lunch programme that gas wholesale prices had not shifted in the way that oil prices had.
He said that if this wholesale price fell, then suppliers were ready to cut bills.
But he added that gas and electricity supplies were bought in advance for the winter, so there would be a lag on any changes in tariffs.
"It is critical that we make sure we have supplies. We are not an energy island," he said.
View from the Commons
MPs joined the debate on Thursday, when Energy Minister Mike O'Brien said he wanted to see the oil price fall feed through into gas and electricity prices.
"If there are benefits we want to see them feed through, not just to individuals who have to pay the bills, but to the economy as a whole to give it a boost," he said.
Conservative energy spokesman Charles Hendry said the government had been slow to offer support for those facing fuel poverty as prices rose.
He also criticised the government for failing to establish a framework for bringing forward investment in new energy-generating capacity or gas storage. This had forced up costs to domestic users and businesses, Mr Hendry said.
For the Liberal Democrats, Steve Webb said that "super-smart meters" should be introduced. These would be able to choose the cheapest energy tariff and supplier for customers automatically.