Fontaine, 89, netted 30 goals from 21 caps for France between 1953-1960.
In 1958, he was instrumental in Les Bleus reaching the semi-finals in Sweden.
A prolific striker with 259 goals from 283 matches in his club career, Fontaine was one of the key players of the great Stade de Reims team who reached the European Cup final in 1959.
Reims, who won three French top flight titles with Fontaine, lost 2-0 to Real Madrid but 'Justo' ended up as the competition's top scorer with 10 goals.
"A star of French football, an outstanding striker, a legendary Reims player... To his family... Stade de Reims send their most sincere condolences," Reims wrote on Twitter.
"A thought for Just Fontaine," French champions Paris St Germain wrote on Twitter.
As a coach, Fontaine helped PSG gain promotion to the first division in 1974.
Brazil beat Sweden 5–2 in the 1958 final in Solna, Stockholm, to claim their first title, having beaten Fontaine's France in the semi-final and Wales in the quarter-final.
The tournament also marked the arrival of a then 17-year-old Pelé on the world stage, who scored in all three of Brazil's knockout games.
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