The government and trade groups have warned of "serious disruption" after France blocked arrivals of UK passengers for 48 hours over concerns about the new coronavirus variant.
Freight lorries cannot cross by sea or through the Eurotunnel and the Port of Dover has closed to outbound traffic.
About 10,000 lorries a day travel between Dover and Calais during peak periods such as Christmas.
UK ministers will discuss the move at a Cobra emergency committee on Monday.
On Sunday, Transport Secretary Grant Shapps urged the public and hauliers not to travel to ports in Kent, saying "significant disruption" was likely in the area.
Kent Police have mobilised Operation Stack - a system to park lorries on the M20 motorway in Kent at times of disruption - to deal with the build-up of traffic.
'Fresh food challenge'
Richard Burnett, head of the Road Haulage Association, told the BBC's Today programme that the ban could deter EU hauliers from coming to the UK over fears they will end up being stranded.
"The retailers have done a good job of stocking up on ambient products [for Christmas] - there will be plenty of stock," he added.
"But the fresh food supply, where it's short shelf life and there will be product on its way now, that's where the challenge comes from.
"The retailers will absolutely be assessing their inbound flows this morning and understanding whether or not those flows are on their way into the retail distribution centres around the country and I'm sure there will be further reassurance given today that those things are in control."
Kent Police said it had implemented the closure of the coast-bound carriageway of the motorway between Junctions 8 and 11 as a "contingency measure".
The Department for Transport has said that Manston Airport in Kent is being readied to take up to 4,000 lorries to ease congestion in the county.
The Port of Dover is closed to traffic leaving the UK "until further notice" due to border restrictions in France, port authorities said in a statement.
"Both accompanied freight and passenger customers are asked not to travel to the port," it said. "We understand that the restrictions will be in place for 48 hours from midnight."
Freight coming to Britain from France will be allowed, but there are fears lorry drivers will not travel to avoid being stuck in the UK.
Unaccompanied freight, such as containers or lorry trailers on their own can still be transported, but outbound vans, lorries and trucks are banned. Hauliers are advised to find other routes into the continent.
Border restrictions could mean disruption to food supplies, as well as difficulties in meeting orders of British goods in continental Europe.
"Tonight's suspension of accompanied freight traffic from the UK to France has the potential to cause serious disruption to UK Christmas fresh food supplies - and exports of UK food and drink," Food and Drink Federation (FDF) chief executive Ian Wright warned on Sunday.
"The government must very urgently persuade the French government to exempt accompanied freight from its ban."
Freight industry lobby group Logistics UK said it was concerned about the welfare of drivers going from the UK to France, and said they should have access to regular testing.
It appealed for calm from shoppers, and said it was "maintaining close contact with UK government to ensure that supplies of fresh produce are available throughout Christmas and the new year".
The British Retail Consortium (BRC) joined the FDF in appealing to the government to find a solution, but also added that there should be no immediate shortages.
"Retailers have stocked up on goods ahead of Christmas which should prevent immediate problems," the BRC said.
The government does not think the restrictions will affect the delivery of Covid-19 vaccines to the UK, according to BBC political correspondent Nick Eardley.
Scotland's First Minister Nicola Sturgeon has called on the government to extend the Brexit transition period as it deals with the new coronavirus variant, saying it was a "profoundly serious situation" which "demands our 100% attention".
The current transition period is due to expire at the end of the year and the EU and UK are still negotiating a trade deal.
Without it both sides will have to collect expensive tariffs that the Office for Budget Responsibility says could harm the UK's economy.
Labour's Rachel Reeves, shadow minister for the Cabinet Office, called the development "deeply worrying".
"The country needs to hear credible plans and reassurance that essential supplies will be safeguarded, including our NHS, supermarkets and manufacturers with crucial supply chains," she said.
The block on freight traffic into France came as a number of European countries banned flights and other travel from the UK over fears about VUI - a mutation of the coronavirus that is spreading rapidly in the UK.
France, Italy, Ireland, Germany, Belgium the Netherlands and Turkey are among those to have banned flights from the UK while other nations are considering the move.