Ministers are working on plans for a temporary visa scheme to make it easier for foreign lorry drivers to come to the UK.
Final details of exactly how the initiative will work are expected to be announced this weekend.
Any changes to immigration rules will be temporary, with a cap on the number of workers allowed to enter the UK.
The shortage of hauliers threatens more disruption to deliveries of petrol, food and other goods.
It is estimated that the UK is short of about 100,000 HGV drivers – with existing shortages made worse by the pandemic and Brexit.
BP and Esso have shut a number of petrol stations due to a lack of tanker drivers, while EG Group has said it will introduce a limit of £30 per customer on all grades of fuel.
The company, which operates 400 filling stations in the UK, said the move was due to the “current unprecedented customer demand for fuel and associated supply challenges”.
All of EG Group’s UK sites remain open and operational to serve customers.
HGV drivers and emergency services will be excluded from the £30 limit due to “their vital role at this time”, according to the firm.
“This is a company decision to ensure all our customers have a fair chance to refuel and to enable our sites to carry on running smoothly,” a spokesperson from the firm said.
“We kindly ask everyone visiting our sites to treat our colleagues, supply chain partners and customers with respect during these very challenging times.”
A Downing Street spokesperson said: “We have ample fuel stocks in this country and the public should be reassured there are no shortages.
“But like countries around the world we are suffering from a temporary Covid-related shortage of drivers needed to move supplies around the country.”
The spokesperson explained that ministers are looking at temporary measures to “avoid any immediate problems”, but added that any measures introduced will be “very strictly time limited”.
“We are moving to a high-wage, high-skilled economy and businesses will need to adapt with more investment in recruitment and training to provide long-term resilience,” the spokesperson added.
The European Road Haulers Association said temporary visas would be a “good idea” but are “only part of the solution”.
The association said it is still easier to drive within the EU than driving between the EU and the UK, meaning that staying within the EU may still be more appealing for drivers.
Despite reassurances, large queues formed at many petrol stations on Friday. On social media, people reported queues lasting for up to 30 minutes, while there were also reports of drivers filling jerry cans.
The AA said that most of the UK’s forecourts were working as they should.
“There is no shortage of fuel and thousands of forecourts are operating normally with just a few suffering temporary supply chain problems,” said AA president Edmund King. “Fridays and the weekend always tend to be busier on forecourts.”