Di Stefano was admitted to the Sagunto hospital, situated just outside Valencia, on Christmas Eve after suffering a heart attack during the night. The 79-year-old was then moved to the intensive care unit at La Fe hospital in Valencia for specialist care.
Di Stefano is said to be in a "serious" condition and needs to improve before he can undergo the bypass operation.
"His heart is beating with difficulty," said a hospital spokesman. "He has a circulation problem and some arteries feeding the heart have been seriously narrowed. He has had a heart attack and operating now would be in unfavourable circumstances."
Di Stefano inspired Real Madrid to win five consecutive European Cups from 1956 to 1960, netting in each final, including a hat-trick in the fifth. He has since been made honorary president of the club.
Di Stefano played international football for both Argentina and Spain and was named European Footballer of the Year on two occasions.