There are likely around 2,000 cases of the new variants in France per day, health minister Olivier Véran said, adding that the number had previously been around 500 earlier this month.
It's one of the several worrying factors in France that could force the government to impose stricter measures, with many anticipating a decision will come soon.
Véran said that France's 6 pm to 6 am curfew was likely having an effect and had probably allowed the country "to avoid the same epidemic curve seen in many neighbouring countries".
But, he added, its effectiveness is "not enough to stop the spread of the virus" especially with the circulation of more transmissible variants.
Earlier in the week, the President of the government's Scientific Council said a national lockdown would likely be necessary to control the spread of the virus.
"If we do not tighten regulations, we will find ourselves in an extremely difficult situation from mid-March," Professor Jean-François Delfraissy said in an appearance on French broadcaster BFMTV, adding that this week was critical in France's fight against COVID-19.
France has reached a plateau of around 20,000 cases a day, a number that has increased around 10% each week for the past three weeks, the health minister said.
Pressure in hospitals is rising with 60% of intensive care units occupied by COVID-19 patients in the country.
Another 250 patients are admitted to intensive care daily with the virus, Véran said. In December, it was 170 patients a day that were being admitted.
There are currently 3,100 people in intensive care and 27,000 people hospitalised in France.
These are similar to the numbers of people who were hospitalised in October when the country decided on a second national lockdown, although then, cases were rising significantly.
The Scientific Council had outlined several potential scenarios in its report published mid-month entitled "A race against the clock."
The possibilities included a national curfew from 6 p.m. with regional lockdowns; a lighter national lockdown similar to the one in place at the end of October; or a strict national lockdown similar to the one in March followed by a curfew.