The prime minister will chair a meeting of the government’s emergency committee later after France closed its border with the UK for 48 hours.
The move, announced on Sunday, means no lorries or ferry passengers will be able to sail from the port of Dover.
France said the action was necessary because of the prevalence of a new variant of coronavirus in the UK.
Nations including Germany, Italy, Belgium, the Irish Republic, Turkey and Canada are suspending UK flights.
Switzerland has also temporarily halted UK flights. Austria is also set to bring in a ban, while Bulgaria has suspended flights to and from the UK from midnight. Unlike the short-term measures in many other nations, its ban lasts until 31 January.
European Union member states are due to meet in Brussels to discuss a co-ordinated response.
Coronavirus cases in the UK rose by 35,928 on Sunday – nearly double the number recorded seven days previously.
And it was announced that further 326 people died within 28 days of testing positive, bringing the nation’s total to 67,401.
Health Secretary Matt Hancock warned that the new variant of the virus – which may be up to 70% more transmissible – is “getting out of control”.
The new variant has spread quickly in London and south-east England, but health officials say there is no evidence that it is more deadly or would react differently to vaccines.
On Sunday, Transport Secretary Grant Shapps urged people not to travel to ports in Kent, saying “significant disruption” was likely.
Kent Police has put Operation Stack into force on the M20 towards Dover to queue lorries caught up in the disruption.
The coast-bound carriageway of the motorway has been shut between Junctions 8 and 11 as a “contingency measure”, the force said.
Manston Airport in Kent is being readied to take up to 4,000 lorries to ease congestion in the county, the Department for Transport has said.
Concerns have been raised by the Food and Drink Federation that the border restrictions could disrupt UK food supplies, and exports of British goods abroad.
But BBC political correspondent Nick Eardley said that the government does not think the travel restrictions will affect the delivery of Covid-19 vaccines, which come from Belgium.
About 10,000 lorries a day travel between Dover and Calais during peak periods such as Christmas.
Freight industry lobby group Logistics UK 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”.
Although freight from France is still allowed to enter Britain, there are fears lorry drivers might not travel because they fear being stuck in the UK over Christmas.
Unaccompanied freight, such as containers or lorry trailers on their own, can still be transported.
Eurotunnel services to France are also suspended and Eurostar trains to Belgium are not operating.
Meanwhile, UK-EU talks to agree a post-Brexit trade deal are set to continue after the two sides missed a deadline for terms set by the European parliament.
MEPs had said they needed to see the terms of any deal by Sunday evening if they are going to be able to ratify it before the end of the Brexit transition period on 31 December.