More than 40 countries have banned UK arrivals because of concerns about the spread of a new variant of coronavirus.
Flights from the UK are being suspended to countries across the world including Spain, India and Hong Kong.
France shut its border with the UK for 48 hours, meaning no lorries or ferries can leave from the port of Dover.
Boris Johnson said he spoke with French President Emmanuel Macron and both sides wanted to resolve "these problems as fast as possible".
The prime minister told a Downing Street press conference: "We had a very good call and we both understand each other's positions."
Johnson, who earlier chaired a meeting of the government's emergency committee, added: "We are working with our friends across the Channel to unblock the flow of trade."
And French Europe Minister Clément Beaune said they would announce on Tuesday what measures were being introduced "after this phase of emergency and harsh precaution that we had to take". He said they would come into effect from Wednesday.
European Union member states met earlier in Brussels to discuss a coordinated response, with officials suggesting a requirement for tests could be imposed on all people arriving from the UK.
It came as the Northern Ireland Executive met on Monday night to consider whether to impose a travel ban between NI and England, Scotland and Wales.
Countries to impose a ban on UK arrivals include Belgium, Canada, Germany, Ireland, Italy, Portugal, Romania, Russia, and Switzerland.
Some of the bans are already in force while others are to begin on Tuesday.
Eurotunnel services to France are also suspended and Eurostar trains to Belgium are not operating.
However, Eurotunnel spokesman John Keefe said he hoped travel from the UK could begin on Wednesday or Thursday.
Meanwhile, Royal Mail says it has temporarily suspended all mail services to Europe, with the exception of the Republic of Ireland, due to the "current restrictions around air, road, ferry and train movements from the UK".
The latest figures released on Monday reveal that another 33,364 people in the UK have tested positive for coronavirus.
There were also a further 215 deaths within 28 days of testing positive, bringing the nation's total to 67,616.
Health Secretary Matt Hancock has 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.
Labour leader Sir Keir Starmer described the situation as a "real emergency", saying: "The news over the last 24 hours has been deeply disturbing. The number of coronavirus cases has nearly doubled in the last week.
"We can have no more over-promising and false hope, confused messages and slow decision-making. We need strong, clear and decisive leadership."
Scotland's First Minister Nicola Sturgeon said the government had plans for coping with port disruption in the event of a no-deal Brexit and should now activate the measures.
Source: BBC
BDST: 1245 HRS, DEC 22, 2020
AP
