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.
As a coach, Fontaine helped PSG gain promotion to the first division in 1974.
Tributes flow in for Fontaine
Fontaine's death has triggered an outpouring of tributes.
"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.
France head coach Didier Deschamps said: "I am saddened by the death of Just Fontaine, as I am sure all those who love football and our national team will be.
"As a player and then as a coach, I was lucky enough to meet him on several occasions. I was lucky enough to meet him on several occasions, notably at his home in Toulouse in September 2017. Justo was a very kind man, very respectful of the generations that succeeded him at Les Bleus.
"His attachment to the French team was strong and sincere. I send a friendly and moving thought to his family, his relatives and all our great elders today in sorrow."
