Di Stefano, who is just over 10 years older than the 76-year-old Bergoglio, wrote in his regular column in sports daily Marca on Monday that the pair had gone to the same school and lived close to each other in the Argentine capital.
"As you can imagine his election filled me with enormous joy," Di Stefano wrote.
"The Pope was probably one of those kids with whom I played football in the street," he added.
"In the neighbourhood we put together proper matches with everyone against everyone until it got dark.
"You'll have to ask him because at that time I was the famous one, from when I was very small, as I belonged to the River Plate youth academy, everyone knew me."
Di Stefano is widely considered one of the greatest players of all time and helped to turn Real Madrid, the La Liga club he joined in 1953, into one of the world's leading sides.
One of Pope Francis's predecessors, John Paul II, who died in 2005, was a goalkeeper in his youth.
- Sports & Recreation