Tara John, Jim Bittermann and Sandrine Amiel, CNN

(CNN) -- France and the UK moved closer to a deal on Tuesday that would allow some personal travel and thousands of stalled freight trucks to resume movement between both countries.

A communique by French Prime Minister Jean Castex's office on Tuesday said French citizens could return to France from the UK starting Wednesday, if they can show proof of a negative Covid-19 test. Other nationals who live in France or the European Union will also be able to enter France from the UK with a negative test result.

Specific procedures for resuming road freight traffic will be detailed in the coming hours, it added.

This arrangement will apply until at least January 6, except if a possible bilateral or European review takes place by then.

"Planes, boats and Eurostars will resume service tomorrow morning. French nationals, residents in France and those who have a legitimate reason must present a negative test," France's Transport Minister Jean-Baptiste Djebbari tweeted.

Fearing the spread of a potentially more contagious coronavirus variant, France was among dozens of countries which closed their borders with the UK over the weekend.

The French closure was the most significant as it also restricted the entry of freight. On Tuesday, nearly 3,000 trucks remained stranded in England and British supermarkets warned of shortages of some goods just days before Christmas.

"Good progress today and agreement with the French Government on borders. We will provide an update on hauliers later this evening, but hauliers must still NOT travel to Kent this evening," UK Transport Secretary Grant Shapps said on Twitter.

The crossing between the English port of Dover and the French city of Calais is a major European trade artery, and the port of Dover handles around 17% of the UK's goods imports.

As of Tuesday morning, about 873 vehicles were waiting at a disused airport site, with approximately 650 vehicles on the M20 motorway, the UK's Department for Transport said.

While drivers from Europe are not prohibited from entering the UK, fears were rising that hauliers would resist doing so for fear of not being able to return home, leading to food shortages.

The period leading up to Christmas is traditionally a busy time for trade, as fresh produce from Europe is imported for the festive period. On top of this, the UK is stockpiling ahead of the end of the Brexit transition period on December 31.

On Monday, Sainsbury's, one of the UK's leading supermarket chains, said If nothing changes, there will be shortages of lettuce, some salad leaves, cauliflowers, broccoli and citrus fruit.