Since 20 December, UK arrivals have been banned from entering the country unless they have a "compelling reason" to visit.
However, according to The Times, an announcement could come as soon as Wednesday (12 January), which would set out plans to lift the restrictions from Saturday (15 January).
Citing French MP Alexandre Holroyd, the paper said the border would open "very soon" to fully vaccinated Brits.
Meanwhile, The Independent reports that some ski operators are so confident restrictions will be eased that they are selling winter sports trips for the coming weekend.