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The heat wave "promises to be exceptional" at the end of June, French meteorological service Météo-France said.
National weather authorities in Belgium, the Czech Republic, France, Germany, Italy, Spain and Switzerland have issued heat warnings, advising people to avoid strenuous activities at midday and in the afternoon, drink plenty of water and stay out of the sun.
Heat records could fall in France
France could see temperatures that exceed those of the historic and lethal 2003 European heat wave. A record-setting mark of 44.1 Celsius (111 degrees Fahrenheit) was measured that August in France, and more than 14,000 mostly elderly people died.
Such temperatures are far above average across Central and Western Europe, hiking the potential danger, CNN senior meteorologist Brandon Miller said.
"When it is 105 (degrees Fahrenheit) in Phoenix or Kuwait, it is not nearly as big of a deal as if it is 105 in Chicago or Paris," he said.
"But when summer temperatures are routinely in the 70s, like in northern Europe or the West Coast of the US, many places do not have air conditioning. This can turn deadly fast if heat waves strike and last for several days.
Mist showers and cooling rooms
Paris City Hall has activated a plan that includes installing 48 mist showers throughout the city, extending hours at public swimming pools, keeping some parks open all night and opening cool rooms in designated public buildings, officials said.
National school exams also have been postponed -- for the first time ever -- to next week, the French education ministry ordered.
"This is the first time this has happened because it's the first time that it's this hot during the school year," education minister Jean-Michel Blanquer said.
Scorching weather in Spain will continue until early next week, with temperatures hitting their highs on Wednesday and Thursday, the Spanish AEMET weather agency warned.
In Germany, where prior heat waves have caused asphalt on highways to bulge, drivers should be extra vigilant, the nation's weather service advised. The country's temperature record for June is likely to fall this week, the agency said.
Neighbors should check on older people during the heat wave, urged Silbernetz, a German old age charity. Older people and children are most at risk of heat-related illnesses.
Children and animals also should be kept away from direct sun and hot cars, the Czech Hydrometeorological Institute warned.
CNN's Celia Heudebourg, Rebecca Blanchard, Antoine Crouin and Saskya Vandoorne contributed reporting.