The World Cup-winning defender made mistakes for both goals as Real lost 2-1 to Manchester City to exit the tournament last month and Varane admits he struggle to make peace with his poor performance.
He told beIN Sports: "Yes, for sure that will not remain a very good memory in my career. It was a difficult game for me, this elimination.
"Immediately after the match I wanted to take responsibility, I made some mistakes, it cost us the victory.
"You had to digest and watch what is happening ahead, and think about what is happening. At the highest level, we don't have time to think about it, we have to think about the next game.
"The days that followed were a bit difficult, I admit, but at the top level, you have to look ahead very quickly."
Varane was also on coy on suggestions that he could succeed defensive partner Sergio Ramos as Real Madrid captain.
The French star said: "Am I ready to take the leadership after Ramos? What is certain is that we are very complementary in our way of playing.
"We have qualities that complement each other perfectly, we have been playing together for several years, so we know each other very well. In defence, it works in pairs and this understanding is very important.
"For the future, after the departure of Ramos, it is difficult to predict things for yourself. I am a leader in my own way. With my team-mates I talk a lot before matches, we talk about tactics, positions. We each have our own way."