Britain, Italy, Switzerland and Spain are also sweltering in extreme heat, with record temperatures in some regions disrupting schools and transport networks.
Europe is warming at more than twice the global average, according to the World Meteorological Organisation, making such prolonged heat episodes increasingly likely.
Much of France is under severe heat alert and set to experience temperatures about 40C on Tuesday, Meteo France said, with temperatures of up to 43C expected in some parts of western France.
The country has just recorded its hottest afternoon and night since records began in 1947, and 54 departments are under red alert in what forecasters said was unprecedented.
Across France, people have been jumping into canals and rivers to cool off.
French sports minister Marina Ferrari said she understood the urge to escape the heat but warned against swimming in unauthorised or dangerous areas.
Speaking before an emergency meeting on the heatwave, French Prime Minister Sebastien Lecornu said: "A sad scourge when it comes to drownings, as the latest figures just reported to us show 40 deaths since June 18, most of them young people."
On Monday, first responders were unable to resuscitate two children, aged two and four, who were found unconscious by their mother in the family car outside their home, authorities said.
In Paris, commuters struggled through sweltering conditions after sleepless nights in apartments ill-equipped for heat.
Some trains were cancelled, including between Paris and Brussels.
Europe's heatwave is driven by a weather pattern known as an Omega block, with a bulge of hot air in the middle and cooler air either side, allowing temperatures to build day after day.
Heatwaves and storms are being intensified by climate change, pushing temperatures higher and causing more rainfall.
Meteo France said conditions were comparable to the August 2003 heatwave, which lasted 16 days and led to an estimated 80,000 excess deaths across Europe, according to the EU.
It was uncertain how long the current episode would last.
In Italy, the health ministry issued its highest level alert for 15 cities and authorities took measures to curtail work in some sectors.
Storms are expected later on Tuesday over the Alps and Apennines, bringing heavy rain, gusty winds and hail.
Britain is also in the grip of the heat, with the Met Office forecasting temperatures of up to 37C in southern England on Tuesday - potentially a June record - before rising further on Wednesday and Thursday.
Dozens of schools plan early closures, citing buildings ill-suited to the heat.
Britain's Network Rail warned passengers to travel only if necessary later this week as temperatures approach 39C, with speed restrictions likely to disrupt services.
Spain's meteorological agency has issued red alerts across parts of the country, warning of dangerous heat with temperatures expected to reach 44C.
Night-time has brought little relief, with about 30 monitoring stations still recording temperatures above 25C early on Tuesday.
Dozens of municipalities across northern Spain cancelled traditional bonfires due to wildfire risks, underlining how extreme temperatures are disrupting both cultural traditions and everyday activities.
In Belgium, soaring temperatures forced a primary school in Tervuren, near Brussels, to relocate its final exams to a nearby church.
In Switzerland, the northeastern canton of St Gallen restricted water withdrawal from rivers and lakes, saying that surface and ground water levels were low and temperatures high.
6°C