The Germany international sustained the injury in the closing stages of the 4-3 win which completed a 5-4 aggregate success over Thomas Tuchel's side.
Liverpool's reward was to be paired with Villarreal in the semi-finals but Can will miss both ties, scheduled for April 28 and May 5.
The club have not ruled him out for the rest of the season, however, and Reds boss Jurgen Klopp was refusing to be downbeat about the 22-year-old's absence.
He said at a news conference: "We have to find other solutions. It's not about moaning about losing Emre.
"It's a ligament in the ankle. It's not the best news but we'll try everything to get him back."