Fuel suppliers say they are struggling to cope with demand for petrol after filling stations saw a week's worth of demand in a single day.
Some filling stations have run dry
Representatives of petrol giants said the issue was one of getting the tankers to the filling stations rather than one of supply.
Drivers are being warned not to panic-buy ahead of fuel protests as queues appear at many petrol stations.
It is feared predictions of shortages may become a self-fulfilling prophecy.
Chris Hunt, director general of the UK Petroleum Industry Association, which represents oil giants Shell, BP, Esso and Total, said tanker deliveries had been boosted by 10 to 15%.
But he said: "We've had a week's-worth of demand in one day.
"I believe folks are still queuing up, but there is absolutely no need to do so."
Total UK said it was "faced with limited product availability at a small number of service stations".
"Our logistics teams are working hard to ensure any disruption and inconvenience to our customers is kept to a minimum."
A spokesman added that the situation was being closely monitored and that Total was working closely with the Department of Trade and Industry on contingency plans in the event of protests.
BP spokeswoman Sheila Williams urged motorists "not to over-buy" but added that some stations had already run dry.
She said deliveries were being made 24 hours a day and that there was "plenty of fuel around".
Esso spokesman David Erlington also urged customers to buy "as normal".
Mark Bradshaw, head of Garagewatch, which represents 6,500 independent retailers in the UK, told BBC News that panic-buying was causing stations across the UK to run dry.
Facts and figures behind UK petrol price rises
But there was no national shortage of fuel, he added.
The Fuel Lobby's Andrew Spence said panic-buying was unnecessary as no attempts to disrupt supplies were planned alongside Wednesday's fuel tax protests.
The farmer from Consett, County Durham, and a prime mover in the 2000 protests, said there would be peaceful protests at refineries around the country.
But he added: "We are not going to restrict any thoroughfare of fuel whatsoever. There will be no blockades."
The Association of Chief Police Officers (Acpo) said any attempts to "disrupt essential goods and services that rely on oil" would be met by "firm action" from police forces.
"Blocking the highway or preventing companies going about their business is unlawful," Acpo said in a statement.
"Chief Constables around the country are currently making arrangements to deal with any incidents."
RAC head of campaigns Sue Nicholson warned panic-buyers could create a repeat of the fuel crisis of 2000 when protests over prices led to a shortage.
"Panic buying is widespread, it's a self-perpetuating, self-feeding frenzy," she said.
"We certainly will face shortages now unless everyone stops trying to squeeze every drop of petrol into their tanks and jerry cans."
Many of the biggest queues are at supermarkets with low fuel prices or in areas with fewer stations - but other places are busier than normal.
The Jet Hartley Service Station in Mannamead Road, Plymouth, sold out of four days' supply in just two days.
Manager Roger Chapman said there had been queues of people buying heavily since Friday.
And one garage owner in Ramsgate, Kent, has stopped selling fuel permanently because he says he is fed up with people complaining about high prices and that he no longer makes enough money to survive.