France says it is working on a plan to "ensure that movement from the UK can resume" - after it banned travel including freight hauliers.
Transport minister Jean-Baptiste Djebbari said cross-EU talks will take place within hours to "establish a solid health protocol" that could begin to unblock the port of Dover.
He added the priority will be "to protect our nationals and our fellow citizens".
Lorries are parked up in Kent with no way to cross the Channel by train or ferry, following the emergence of a new strain of coronavirus that Boris Johnson said could be up to 70% more transmissible than existing ones.
The prime minister will hold a Downing Street news conference on Monday afternoon, and before that chair a meeting of the government's emergency committee known as COBRA in a bid to avert pre-Christmas shortages.
On the agenda for ministers to discuss is the supply of food and medicines, traffic management, border controls and vaccines.
They will also talk about engagement with EU countries and the local impact on ports, as well as testing at the border.
Tim Rycroft of the Food and Drink Federation said there was "no reason to be concerned for shoppers at the moment" but voiced hopes the current serious disruption is a "wake up call to remind us of the fragility of these supply chains".
A growing number of countries are banning travel from the UK due to fears about the new coronavirus strain.
France went one step further by also restricting hauliers for 48 hours - while most others have exempted them to keep accompanied trade moving.
Mr Djebbari defended the decision, writing on Twitter that "every time" the French borders are opened "we are told that we are killing people", while "whenever a country does a little more than us, we are told that we are lax".
And the country's Europe minister added: "We were never among those who said the virus stopped at the borders. We're pragmatic, we're protective. We had to take a precautionary and emergency measure."
The seriousness of the new coronavirus strain was underlined by Prof Andrew Hayward, a scientist who sits on the UK government's New and Emerging Respiratory Virus Threats Advisory Group.
"If a vaccine was the best news - and it still is in this pandemic - then this is the worst news we've had so far," he told Sky News. "We really need to tighten down the hatches in order to stop the spread of this strain."
Prof Hayward added the UK "should be taking our own action to stop international travel - that would be the responsible thing to do".
There have been record high numbers of daily COVID-19 cases in recent days, leading to the Christmas rules relaxation being significantly toughened in most parts of England and scrapped altogether in Tier 4 areas.
Earlier, the UK's Transport Secretary Grant Shapps insisted travel chaos would not impact distribution of the COVID-19 vaccine, which is being imported from Belgium.
Watch our special programme tonight as Isabel Webster is joined by a panel of experts to answer your questions about the new strain of the virus, COVID Christmas rules and the new Tier 4 measures.
COVID Christmas Crisis Q&A is at 6.30pm.
Send us your questions via WhatsApp on the number 07583 000853.