More than 2,850 lorries are stuck in Kent waiting to leave the UK, the county council has said, as politicians thrash out a plan to reopen France's border to trade and travel.
France shut its UK border for 48 hours on Sunday amid fears of a new coronavirus variant.
Any solution would probably include testing for lorry drivers, BBC Paris correspondent Hugh Schofield said.
Meanwhile, the EU Commission has urged countries to drop their travel bans.
More than 50 countries have banned UK arrivals following widespread concern about the spread of the new variant.
In a recommendation to all member states, the commission said flight and train bans should be discontinued to avoid supply chain disruption.
People should be allowed to travel to their country of residence, provided they take a Covid-19 test or self-isolate, it said. But the commission added that non-essential travel should still be discouraged.
Currently, no lorries are leaving the Port of Dover or Eurotunnel to France.
The leader of Kent County Council, Roger Gough, told the BBC that 2,220 vehicles were at the temporary lorry park at Manston, while 632 were still being held on the M20.
It comes as Tesco said it would be reintroducing temporary purchasing limits on some essential products, including toilet rolls, eggs, rice and hand wash.
A further 36,804 people in the UK have tested positive for coronavirus and there were 691 deaths within 28 days of a positive test, according to Tuesday's government figures.
It is the largest daily number of cases recorded yet, though it is thought the infection rate was higher during the first peak in spring when testing was much more limited.
Earlier, Home Secretary Priti Patel told BBC Breakfast discussions were under way between the UK and France "to find a resolution" to the Channel disruption, with an update on the situation expected later.
Prime Minister Boris Johnson and French President Emmanuel Macron spoke on Monday and, according to the BBC's Hugh Schofield, a solution to the blockage will probably include compulsory negative Covid tests for lorry drivers coming into France from the UK.
Any plans agreed between the leaders would come into effect on Wednesday, France's Europe minister, Clément Beaune, said.
Ms Patel said potentially testing lorry drivers at ports was "part of the discussions", adding that setting up testing for this could "happen relatively quickly".
EU member states are understood to be pressing for UK arrivals to be tested for the virus before entering their countries.
Bosses at Eurotunnel estimate that up to 2,500 freight vehicles are expected to arrive in the UK on Tuesday, and the same again on Wednesday.
The port of Harwich in Essex - which is about 130 miles from Dover by road - is also seeing a build up of lorries as drivers divert from Dover, a spokesman for the port said. The port of Felixstowe in Suffolk is also busy but there are no queues.
The British Retail Consortium said trucks needed to be able to start travelling again in the next 24 hours to "avoid seeing problems on our shelves".
Andrew Opie, its director of food and sustainability, told the Commons business, energy and industrial strategy committee: "What we've been told by members is that unless those trucks can start travelling again and go back to Spain and Portugal and other parts of Europe, we will have problems with fresh produce from 27 December."
Labour's Rachel Reeves told BBC Breakfast the government had "wasted the last 24 hours" and called for "testing in place so we can reopen the border".
In other developments:
- The company behind the Covid-19 vaccine being given to people in the UK says it is "highly likely" it will protect people against the new variant of the virus
- Northern Ireland's Executive votes against introducing a travel ban between Northern Ireland and England, Scotland and Wales
- Royal Mail said it had temporarily suspended all mail services to Europe, with the exception of the Republic of Ireland, due to travel restrictions
- The number of excess deaths in the UK since the start of the pandemic has passed 81,000
- Army soldiers have been drafted in to drive ambulances in Wales
The border disruption has also affected passenger services - with many air, rail and sea services cancelled between the UK and France, as well as other countries.
Eurotunnel said it hoped passengers would be able to travel between the UK and France from Wednesday or Thursday, if a solution was agreed.
British Airways said it would operate "a reduced and dynamic schedule" amid the uncertainty.
The Channel is a vital trade route, with about 10,000 lorries a day travelling between Dover and Calais at Christmas, largely bringing in the freshest produce.
Shane Brennan, the head of the Cold Chain Federation which represents the UK's temperature-controlled transport industry, said the number of vehicles stuck was worse than the government figures suggested, as lorries were "dispersed around southern England".
Ms Patel said the numbers do "fluctuate", and added there were welfare facilities and support available for hauliers at Manston Airport.
Queues on the M20 in Kent aren't exactly unheard of, but the French decision to close the border is dramatic and has caused a lot of disruption.
Politicians on both sides of the Channel are hopeful they might be able to agree a way of getting things moving again before the end of the year.
But if that requires a massive expansion of testing for coronavirus at the border, that's easy to say, far harder to do.
In England, 17 million people are under tier four rules, the toughest level, where people are being told to stay at home and not leave the area. Some parts of England, as well as Wales, have asked people who travel from tier four areas to self-isolate.
Wales has entered a new national lockdown while Scotland has said it might need to tighten its top level of rules (level four), ahead of the whole nation moving into that level on Boxing Day. Northern Ireland will begin a national lockdown on Boxing Day.
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