Di Stefano helped Real dominate European football in the 1950s
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Real Madrid great Alfredo di Stefano will undergo surgery on Wednesday if his condition remains stable.
Di Stefano, 79, suffered a severe heart attack on Christmas Eve and reached critical point on Sunday before improving slightly.
"In the last 24 hours, and though the prognosis remains serious, there's been an improvement in his circulation," Valencia's La Fe hospital said.
"If the stability continues, the bypass operation will take place tomorrow."
A hospital spokesman said on Monday: "It is a complex case because he is a diabetic and is 79 years old, but despite that I'm optimistic and I think there is a good possibility of him making a recovery.
"He is conscious, aware of what is going on and doesn't need any artificial respiration. He is a strong man and it is clear that he has a great deal of inner belief."
Di Stefano helped Real win five European Cups and eight league titles and was twice European player of the year.
He was born in Argentina and played for his homeland, Colombia and Spain - but never featured in the World Cup finals.
Di Stefano began his career with Argentine giants River Plate in 1945. He then moved to Colombia's Millionarios in 1949, where he won four league titles and scored 267 goals in 292 games.
He joined Real Madrid, after a tug-of-war with rivals Barcelona, in 1953 and was instrumental in their amazing five consecutive European Cup titles from 1956.
Di Stefano scored goals in every final, including a hat-trick in their famous 7-3 victory over Eintracht Frankfurt at Hampden Park in 1960.
He held the record as top scorer in the European Cup, with 49 goals in 58 games, until Real Madrid striker Raul beat his mark this season.
He was voted European Footballer of the Year in 1957 and 1959.
Coaching success
He left Real for Espanyol after falling out with club president Santiago Bernabeu in 1964 and retired three years later, playing his final match against Celtic in Madrid.
He then went into coaching, leading Boca Juniors and River Plate to Argentine championships and steering Valencia to a Spanish league title in 1971 as well as victory over Arsenal in the European Cup Winners' Cup in 1980.
He returned to Real Madrid to take over as coach between 1982 and 1984, with the side finishing runners-up in the title race in consecutive seasons.
Di Stefano was made honorary president of Real in 2000 when Florentino Perez came to power.