The row about high energy prices has been ignited after British Gas announced its profits had doubled in the first half of this year.
It comes as its parent group Centrica said it would be axing 4,000 jobs, not all of which will be in the UK.
The cut follows a strategic review aimed at delivering savings of £750 million a year by 2020.
British Gas profits were £528 million for the first half of the year, more than it made in the whole of 2014, even though it recently cut prices by 5%.
Which? executive director Richard Lloyd said: "While it's good to see the new commitment from Centrica to focus on serving customers, with British Gas profits high and wholesale prices low, customers will no doubt wonder why cuts to their bills haven't gone further, and haven't included electricity.
"Following the Competition and Markets Authority's blistering assessment of this sector, we expect big suppliers to pass on falling costs to their customers quickly and fairly."
Centrica is to axe thousands of jobs following a strategic review, it was announced today.
The British Gas owner said it was reducing its headcount by 6,000 across the group, not all in the UK.
But the company will create 2,000 other jobs so the net effect will be 4,000 job losses.
British Gas' operating profit was up by 44%, according to today's interim first half results from the group, which was larger than expected.
The company said this was because of a colder year compared to 2014, with consumption up by 11%.
Energy bills are likely to go up in the longer run if Labour come to power, the chief executive of British Gas owner Centrica has told ITV News.
Leader Ed Miliband pledged to freeze energy bills for 20 months if his party wins the general election, but Sam Laidlaw says the plans are likely to scare investors, thereby raising the cost of capital for energy companies and in turn leading to higher bills.
Asked if the plans were already having such an effect, Mr Laidlaw said general "political uncertainty" was raising the cost of capital for energy firms.
The boss of British Gas parent Centrica has told ITV News the company will make far less profit from energy bills than watchdog Ofgem has suggested.
Chief executive Sam Laidlaw told ITV News Business Editor Joel Hills the energy firm would make £40 profit per customer per year - some way from the £106 Ofgem says energy companies are set to make.
He added the watchdog's estimates were "not an accurate representation" and do not "help consumer trust".
A milder winter and energy efficiencies helped British Gas profits tumble by 26% in the first half of 2014.
Parent company Centrica predicts average bills will fall by around £90 this year because consumers are not using as much power.
However, industry regulator Ofgem has suggested that it sees power companies doubling their profit margins in the second half of the year.
They have already referred the Big Six energy companies to the competition authorities.
ITV News Correspondent Nina Nannar reports:
Online energy comparison site U-Switch says despite a fall in profits, British Gas should be passing on savings to customers.
Anne Robinson from U-Switch says the 26 per cent fall is based on the weather and it was time the energy giant passed on the low wholesale prices to its customers.
She added it was clear "loyalty doesn't pay" and consumers could save as much as £200 a year by shopping around.
British Gas energy prices could come down before next year's General Election, the boss of parent company Centrica has suggested.
Speaking to BBC Radio 4's Today programme, chief executive Sam Laidlaw said: "I think we have a strong record of being the first company to reduce prices wherever we possibly can and, if we see the opportunity, we would do so before the election or after the election."
Mr Laidlaw was pressed about the impact of Labour's pledge to freeze prices but said the costs Centrica incurred would be the key factor in deciding whether or not there could be any reduction.
He said the company's drop in profits was related to two "unusual" weather phenomena - the warmer weather in the UK and the US polar vortex which "resulted in generating companies charging us with a lot of additional sort of ancillary costs".
British Gas owner Centrica has come under pressure to cut bills after watchdog Ofgem pointed to falling wholesale gas and electricity prices.
Customers are paying higher tariffs this year after they were hiked 9.2% last autumn, though the rise was scaled back following the Government's shake-up of so-called green levies on bills.
But Centrica said the average bill was expected to be £90, or 7%, lower this year reflecting warmer weather and energy efficiency measures.
The company reiterated that tariffs were not expected to change during 2014 "recognising competitive conditions in the UK energy supply market".
Centrica has defended its stance by pointing to the way it buys much of its energy in advance and that the benefit of lower wholesale prices for next year is offset by higher costs elsewhere.
British Gas owner Centrica has reported pre-tax profits of £890 million for the first half of 2014, a drop of 40% from 2013.
The company said the fall reflected a changing market environment and the effects of mild weather in the UK and the polar vortex in North America.
ITV News Business Editor Joel Hills tweeted:
Mild weather in UK and polar vortex in US dents Centrica's profits. Owner of British Gas made H1 pre-tax profit of £890m - down 40%.