Originally slated to start from Nice on June 27 and finish in Paris on July 19, there is no chance the event can go ahead as planned and organisers face a mammoth logistical task of rescheduling.
So far, organisers ASO have remained silent, but several mayors of the French towns along the planned route say they have been consulted regarding the new dates.
French President Emmanuel Macron said in a televised address Monday that a strict lockdown in France would continue until at least May 11, as public gatherings were banned until mid-July.
Tour general director Christian Prudhomme has said riders will need two months after the lockdown ends to prepare for the race.
France's Minister of the Interior Christophe Castaner said on Tuesday that ASO had to reschedule or cancel the Tour.
"It is up to the organiser to analyse their ability to organise it and reschedule it," Castaner told French radio.
A start in late July or mid-August has been mooted, with some reports suggesting the race could even be delayed until September.
The race's route is over 3,000km long, with roughly 500,000 fans lining the roads each day.
"Social distancing on the roadside wouldn't be a problem, but in the start towns, at the finish line and in the VIP tents it certainly would," Belgian virologist Marc Van Ranst told Flemish TV channel Sporza on Saturday.
Suggestions the Tour could be held without supporters have been ruled out.
"The Tour de France is 3000km of smiles," Prudhomme said regarding roadside gatherings. "We won't run a Tour de France without the fans."
- Between a rock and a hard place -
Macron's announcement may have caught the organisers short.
A late July start for the Tour could be considered too close to the ban on large public gatherings, while the riders would also need to be in peak condition.
ASO also organises the Criterium du Dauphine eight-day race, and had been hoping to run that ride through the Alps in late June or early July, but the traditional Tour warm-up has been postponed.
A mid-August start would see the race finish in September, meaning a clash with the Vuelta a Espana.
There are only 176 riders on a Tour, but the whole event involves around 4,500 people, with team staff, police and media all moving every day.
Cancellation is the worst-case scenario.
The "Grande Boucle", as the Tour is known in France, is the central economic pillar which supports the sports' 22 professional teams.
"It's as simple as this. If the Tour does not take place, teams could disappear, riders and staff alike would find themselves unemployed," said Marc Madiot, the chief of French team Groupama-FDJ.
The riders are also keen to race, with 2018 winner Geraint Thomas saying he was desperate for the Tour to go ahead.
"There are bigger and more important things to sort out first, but as soon as it's safe and ready to go, we (Ineos) would love it to happen," said the Welsh rider.