Morrisons is set to cut the cost of diesel and unleaded by up to two pence per litre.
The supermarket said it is "doing its best to pass on the benefit of lower fuel prices following favourable changes in the value of sterling". Morrisons has 333 filling stations across the UK.
The pound has enjoyed a week of post-Brexit vote highs, although it tumbled this morning ahead of Theresa May's hotly anticipated speech in Florence.
"We are responding quickly and we are cutting both diesel and unleaded by up to 2p a litre," said Steve Mosey, Morrisons services director.
RAC fuel spokesman Simon Williams said: "This is some very welcome relief for motorists who have suffered a 3p a litre rise in the average price of petrol since mid-August as a result of Storm Harvey hitting the US Gulf Coast and disrupting refineries.
"Motorists are now paying nearly 120p a litre for unleaded – the highest since March – while the average price at the big four supermarkets is 117p. This latest cut from the supermarkets should force forecourt reductions across the country, hopefully bringing prices down by 2p a litre everywhere.
"Strangely, this has all come at time when the oil price has increased by around $7 a barrel since mid-August, but it is the strengthening of the pound to $1.35 that has made the big difference as fuel like oil is traded in dollars."