Los Blancos needed a victory to secure a spot in the quarter-finals but ended up losing 2-1 at the Etihad on Friday evening, as they bowed out at the last 16 stage for a second season running.
With Sergio Ramos suspended, Varane had a night to forget as City's two goals on the night were as direct result of his two errorrs.
"This defeat is mine, I have to accept it," he told reporters, as per Goal.
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"I am responsible for this defeat. We had prepared well but you pay for your mistakes. I am sad for my team-mates.
"At 1-1 we had the chance to qualify and today things did not go to plan.