Energy suppliers risk "undermining public confidence" in the market by failing to respond to falling wholesale prices, the regulator has said.
Ofgem said the "big six" gas and electricity companies should explain to customers why falling wholesale prices have not been passed on to household bills.
Under one measure, wholesale prices are at their lowest level for four years.
Wholesale costs make up nearly half of a household energy bill.
British Gas, the UK's largest supplier, said this meant the wholesale price could fall, but this might not push down prices overall.
"Our recent trading update demonstrated that these reductions in wholesale costs have most definitely not translated to increased profits for British Gas," a British Gas spokeswoman added.
The big domestic suppliers tend to buy energy in advance, in what is known as the forward market, in an attempt to avoid any short-term rises in costs.
The forward prices for the coming winter are 16% lower for gas and 9% lower for electricity than last winter, Ofgem said. The mild winter in the UK and Europe has left record levels of gas in storage, pushing down prices.
Spot prices - when firms buy energy for next day delivery - have fallen even more sharply. These tend to be used by smaller suppliers who have more flexibility owing to their smaller market share.
Spot gas prices were down 38% in early June compared with the same time a year ago. This left prices at their lowest level since September 2010. Spot electricity prices were 23% lower than a year ago, taking them to their lowest level since April 2010.
In a competitive market, a sustained reduction in costs would encourage suppliers to cut prices for customers, as they would be worried about losing customers to other suppliers, Ofgem said.
However, the regulator said falling wholesale costs had not been reflected in falling household bills.