Maradona is widely regarded as one of the greatest footballers of all time and was the inspiration for Argentina's World Cup success in Mexico in 1986, winning the Golden Ball as the tournament's best player.
The Argentine FA confirmed the news in a statement on Twitter which read: "The Argentine Football Association, through its President Claudio Tapia, expresses its deepest sorrow for the death of our legend, Diego Armando Maradona.
However, he remained a revered figure at the Italian club, where he won two Serie A titles. He also played for Barcelona, Sevilla, Boca Juniors and Newell's Old Boys.
The 1986 World Cup winner had an emergency operation for a subdural hematoma, which is an accumulation of blood between a membrane and his brain.
Maradona was initially admitted to another clinic in La Plata with signs of depression, anemia and dehydration, before being moved to Olivos when the subdural hematoma was discovered.
Dr Luque said an accident likely caused the subdural hematoma but that Maradona did not recall any falls or mishaps.
Manchester United and England forward Marcus Rashford paid tribute on Twitter, while Gary Lineker wrote: "By some distance the best player of my generation and arguably the greatest of all time. After a blessed but troubled life, hopefully he'll finally find some comfort in the hands of God. #RipDiego"