A source told the Daily Telegraph the measure was discussed at a meeting on Wednesday as part of the review of international travel by officials.
The newspaper added that ministers have agreed to monitor the situation.
Should France be moved to the “red list”, all arrivals coming from there would be required to quarantine in a hotel for 10 days upon arrival in the UK.
The Beta coronavirus variant was first identified in South Africa, and is thought to be more resistant to vaccines than other strains of the disease.
France is currently in the medium-risk amber list of countries.
Earlier this week, Scotland’s National Clinical Director, Professor Jason Leitch said he hopes changes to travel regulations, which mean those who have been fully vaccinated no longer need to isolate after returning home from amber-list countries, could encourage more people to be inoculated.
The number of vaccinations reached a three-month low earlier this week, with 17,749 injections – including 7,163 first doses – administered on Monday.
Speaking on BBC Radio Scotland’s Good Morning Scotland programme on Thursday, Prof Leitch said: “If travel is an incentive for you, here’s another reason, if you needed another reason other than the illness, to get yourself vaccinated.”
He urged people to “please, please, please get yourself vaccinated”.