One of the greatest footballers of all time, Argentina-born Di Stefano was taken to the intensive care unit of Madrid's Gregorio Maranon hospital after an 18-minute respiratory arrest.
The latest medical report, which Madrid published on their official website on Sunday, read: "The patient, Alfredo Di Stefano, 88-years-old, admitted to the coronary care unit, remains in the Cardiology Department at the Gregorio Maranon Hospital after suffering a cardiac arrest.
"The patient remains clinically and haemodynamically stable, but his condition remains serious."
According to reports in Spain, Di Stefano was on a street close to the Bernabeu when he had the heart attack - a day after his 88th birthday. The hospital said the emergency services were able to stabilise him.
Madrid confirmed on Saturday evening that Di Stefano was under sedation, with endotracheal intubation and artificial ventilation, and was being subjected to the relevant clinical trials.
Known as 'La Saeta Rubia' (the blond arrow), he has had several health problems in recent years and is often seen in a wheelchair in public.
Di Stefano was fitted with a pacemaker in 2005 following heart surgery.
His achievements as a player helped turn Real, the club he joined in 1953, into one of the world's leading sides.
The skilful Di Stefano featured in attack when the Madrid team won five successive European Cups between 1956-60 and he scored in each of the finals.
Former Real striker Gonzalo Higuain, whose goal sent Argentina into the World Cup semi-finals in Brazil on Saturday, said: "I was told when I got to the dressing room. It is not good news for football, for those who know him, for those who are close to him.
"Alfredo was always very close to me, I feel for him, I have great respect for him and I hope that he will recover. I only have words of gratitude towards him."
Real's official website carried a message of good wishes and picture of club president Florentino Perez visiting Di Stefano at the hospital on Saturday.
Di Stefano also won four Colombian and two Argentine league titles as a player.
- Sports & Recreation
- Di Stefano
- Gregorio Maranon