The 22-year-old, who joined Wolfsburg from Schalke in August, impressed in the first leg of the quarter-final tie, but limped off at the half-hour mark in the return at the Santiago Bernabeu, which saw Madrid progress to the last four courtesy of a 3-2 aggregate win.
Draxler underwent further examination on his injury on Thursday and Wolfsburg have now announced he faces a lengthy spell on the sidelines.
"I am disappointed and it's a real shame that I can't help the team right now in looking to achieve our goals for the season," Draxler said.
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"I will now do everything within my power to get back into action again as quickly as possible."
Head coach Dieter Hecking, meanwhile, has acknowledged Draxler's absence is a major blow to the team, but he is hopeful they can cope without the gifted midfielder.
"That is a bitter blow for Julian and VfL," Hecking said.
"We wish him all the best for a problem-free recovery and hope he is fully fit again soon.
"Right now, we have to focus on quickly compensating the loss. We have good options in midfield and will find a solution."