The Spain international, who has underperformed this season and has not scored in La Liga since September, picked up the injury in Wednesday's 3-1 win over Kashima Antlers in the semi-finals.
He came on as a substitute and then had to be replaced 14 minutes later.
"After tests carried out on our player Marco Asensio by the medical staff, he has been diagnosed with a grade two injury in the quadriceps in his right leg," said a medical report from Real on Friday.
The report did not state for how long Asensio would be on the sidelines, although local media reported he would be out for around a month.
(Reporting by Richard Martin; Editing by Toby Davis)
