But EU officials could be ready to offer a reprieve as they consult member states on the possibility.
A positive decision could be taken by Brussels in a matter of weeks, according to the Financial Times.
The green card system operates in a 48-country zone that includes the entire EU and UK.
Within the bloc, motorists are legally allowed to drive without the insurance paperwork.
Without the exemption, drivers in the green card zone who don’t carry proof of third-party insurance are liable to be stopped or even have their vehicles impounded by foreign police forces.
Brussels refused to consider offering Britain a waiver while they were locked in the turbulent post-Brexit trade talks.
But with the future relationship pact now in place, eurocrats are more likely to recommend that EU nations accept British drivers without insurance documents.
Graeme Trudgill, executive director of the British Insurance Brokers’ Association, said the green card waiver would come as a welcome boost for motorists ahead of the summer holiday season.
He said: “This waiver is something we have pushed for long and hard.
“If this is confirmed it would be fantastic news for British motorists.”
But eurocrats left Britain’s request to join the scheme completely unanswered during the whole of 2020.
Britain’s application is being considered alongside that of Montenegro.
EU officials have said Brussels was working with national governments on a plan to admit the countries, with capitals still needing to confirm the final details before the requests are taken forward.