A temporary visa scheme to make it easier for foreign lorry drivers to work in the UK is to run for three months, ending on Christmas eve.
It is understood that about 5,000 temporary visas could be issued, with more details expected on Sunday.
A shortage of drivers has disrupted fuel deliveries, with some petrol stations closing, and queues forming.
The government said there was no fuel shortage at refineries, and urged people not to panic buy.
The Road Haulage Association estimates that the UK is short of about 100,000 HGV drivers – with existing shortages made worse by a number of factors, including the pandemic and Brexit.
Business minister Greg Hands said fewer than 1% of petrol stations were closed and there was “easily enough petrol available for normal use at this time of year”.
He told BBC Radio 4’s Any Questions programme the UK leaving the EU meant the government could make it easier to become an HGV driver and help make the profession more attractive through higher wages.
MPs and police forces have called on drivers to be “sensible”, to consider filling up another time, and to avoid stockpiling fuel in jerry cans.
One petrol station boss from Ferndale, in Rhondda Cynon Taf, appealed for calm amid continued high demand for fuel on Saturday.
Meanwhile, the continued shortage of hauliers threatens more disruption to deliveries of petrol, food and other goods.
A raft of longer-term measures including training up more drivers and addressing the backlog of driving tests are to be introduced by ministers.
The AA said current queues at petrol stations were unlikely to last because the supply chain was not hit by ongoing problems.
“The good news is you can only really fill up once – you’ve got to use the fuel, so this should be a short-term thing,” AA president Edmund King said.
“So, once people have filled up, they won’t travel more than they normally travel, so this strain on the system should ease up in the next few days.”
Conservative MP Tobias Ellwood welcomed the introduction of the temporary visa but said this would not help overnight.
As well as training more domestic drivers and speeding-up testing, Mr Ellwood outlined other “imaginative ideas”.
“We could retrain hundreds of Afghan refugees many of them are over here, they drove much bigger HGVs in their own countries,” he said. “There are solutions out there but we need to act.”
Some drivers who spoke to the BBC on Saturday gave differing views of the situation.
Rosie Osborne, a wedding singer, is queuing for petrol in Balham Hill, south London, before travelling to a wedding in Somerset, where she will front the band.
“I’m in the red and I need to get petrol ASAP,” she says. “I have been sat in this queue for 50 minutes and I didn’t want to risk getting petrol elsewhere en route, but I am nearly on empty so I need to get petrol anyway.”
But there was no such pressure for one driver in Leeds who says that, while she only has 21 miles left in her tank, “I’ll just walk or get the bus” if she can’t fill up.
One woman, who lives in York, says she had managed to get some fuel on Friday night which she needed to enable her to help her 91-year-old mother who lived in Leeds.
“She would have been stuck,” she says. “It’s very difficult.”
Another driver questions whether the cost of the training required to become a licensed heavy goods driver was off putting.
“That maybe why there’s an issue – it’s not an industry people want to get into because it’s so expensive to get into,” she says.
Sainsbury’s said it was experiencing “high demand” for fuel, with a “tiny proportion of sites” temporarily closed.
BP said about 20 of its 1,200 petrol forecourts were closed, with between 50 and 100 sites affected by the loss of at least one grade of fuel.
A “small number” of Tesco refilling stations have also been impacted, said Esso owner ExxonMobil, which runs the sites.
EG Group, which has 341 petrol stations in the UK, is introducing a limit of £30 per customer on all grades of fuel due to “unprecedented customer demand”.
One petrol station in Stockport sold 5,280 gallons (24,000 litres) of fuel on Friday, compared with 1,760 gallons (8,000 litres) on the same day the previous week.