Canada and India joined European states in blocking flights from the UK while Europe-bound train services via the Channel Tunnel have been halted.
The new variant is said to be up to 70% more transmissible, but there is no evidence that it is more deadly.
There is also no proof to suggest that it reacts differently to vaccines.
Two meetings are taking place in Brussels on Monday - one involving health ministers and another with the EU's crisis response team. But no decision is expected until Tuesday, when EU ambassadors meet.
A French official told the BBC's Gavin Lee that they were desperate to reopen the borders "as soon as safely possible", with one option discussed being the requirement that UK travellers - including lorry drivers - prove they have had a recent negative Covid-19 test.
UK Prime Minister Boris Johnson said he had an "excellent" conversation with French President Emmanuel Macron and was keen to find a solution "in a few hours".
"I want to stress we in the UK fully understand our friends' anxieties about the new variant. But it's also true that the risks of transmission sitting alone in the cab is very low. So we hope to make progress," he said.
UK Health Secretary Matt Hancock earlier said the new variant was "out of control", while Denmark, Italy and the Netherlands announced they had already detected it.
Following the discovery of case in Denmark, neighbouring Sweden banned all foreign travellers from there.
Also on Monday, the EU's medicines regulator approved the Pfizer-BioNTech coronavirus vaccine for use in all 27 member states.
The vaccine is already being administered in the UK and in the US, and some European countries plan to begin administering doses from 27 December.