Talks with France on resuming full trade and transport across the Channel were continuing on Tuesday amid warnings that the border must be running again by Wednesday to avoid disruption to food supplies.
Home Secretary Priti Patel said the Government is "speaking constantly" with France to achieve a resolution "in both our interests" to get freight moving again.
More than 40 countries have banned flights from the UK due to a mutant variant of coronavirus spreading through the country, while lorry drivers spent a second night sleeping in their cabs on the M20 outside the Port of Dover, which has been shut since Sunday night.
Reports suggest a possible solution could be mass testing of HGV drivers across the Channel, while the BBC reported that plans to reopen the border will come into effect from Wednesday, citing French Europe Minister Clement Beaune.
Andrew Opie, director of food and sustainability at the British Retail Consortium, said the "borders really need to be running pretty much freely from tomorrow to assure us that there won't be any disruption".
He told BBC Radio 4's Today programme: "There is a problem potentially directly after Christmas and that is really in fresh produce, so we're talking here about things like salad, vegetables, fresh fruit, of which the vast majority come from Europe at this time.
"The problem actually is empty lorries, so the empty lorries which are now stuck in Kent, they need to get back to places like Spain to pick up the next consignment of raspberries and strawberries and they need to get back within the next day or so, otherwise we will see disruption."
He added: "As long as it can be cleared today, there'll be minimal impact for consumers – remember the shops are shut on Christmas Day which takes one day of buying out of the equation, but those lorries that are stuck in Kent, they do need to get back within the next day."
Ms Patel told Sky News: "We speak to our colleagues in France constantly on a range of issues and that work has been under way over the last 24 hours and will continue today.
"We're working to get a resolution, I think that's really important to put this into context. It's in both our interests, both countries to ensure that we have flow, and of course there are European hauliers right now who want to be going home and, quite frankly, it's in both our interests to carry on those discussions and negotiations and we will see what materialises today."
Government sources told the PA news agency that discussions with the French government are "ongoing" after Prime Minister Boris Johnson said on Monday evening that the two countries are working "to unblock the flow of trade as fast as possible".
The M20 in Kent was closed on Monday night to allow for the implementation of Operation Brock – contingency measures which involve using a moveable barrier to keep traffic moving on the motorway whenever there is disruption at the Channel.
Meanwhile, the Government's chief scientific adviser, Sir Patrick Vallance, said tougher restrictions may be needed across the UK because cases of the mutant strain of Covid-19 are now appearing "everywhere" and it is transmitted more easily.
London and parts of southern and eastern England were rushed into the new Tier 4 lockdown regime at the weekend, effectively cancelling Christmas plans and imposing measures similar to the previous national lockdowns.
At a Downing Street news conference on Monday, Sir Patrick said: "I think it is likely that this will grow in numbers of the variant across the country, and I think it's likely, therefore, that measures will need to be increased in some places, in due course, not reduced."
Scotland and Northern Ireland have already announced new lockdowns from Boxing Day, while Wales's tough restrictions will only be eased for Christmas Day before being reimposed.
Northern Ireland's Executive voted against imposing a travel ban on Great Britain and the Republic of Ireland overnight but did agree to issue guidance warning against all but essential travel.
But more positive news came from the Prime Minister, who said more than 500,000 people have now received their first dose of the Pfizer/BioNTech Covid-19 vaccine.