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Juve have dominated Serie A with eight straight title wins, but their last success in Europe's premier club competition came in 1996.
The Italian giants were in the last 16 this season - losing the first leg of their tie away at Lyon 1-0 - when the campaign was suspended due to the coronavirus pandemic.
Khedira, who arrived at Juventus in 2015, said he wanted to help deliver the Champions League trophy to Turin.
"I won almost everything, but winning is like a drug, if you don't win a title for one year, the next year you try to win all kinds of titles," he said on Instagram on Thursday.
"My dream is to win the Champions League with Juventus. Almost three or four years ago, we were very close in the final against Real Madrid.
"I think we can win this title with Juventus in the next few years. In my opinion, this is not just my dream, but also that of every Juventini."
Khedira helped Madrid win the Champions League in 2014 among numerous other trophies with the La Liga giants.
A World Cup winner with Germany in the same year, the midfielder remembers his time in Madrid fondly.
"It was awesome. I spent unforgettable moments there with many successes. We won all sorts of titles like the league, the cup, but also the Champions League," Khedira said.
"We had a very strong team with very strong characters. That's why I will never forget this great time and I am still a big fan of this big club. That's why: Hala Madrid!"
With his current contract with Juventus set to expire next year, Khedira recently stressed he's not looking at moving on anytime soon.
"My contract with Juventus expires in 2021. I feel very good in Italy and in the team," he said. "I have found a very strong team and I am still hungry to win many titles.
"Now I have more confidence in my body and therefore I see no reason to change team. I am very happy at Juventus and in Turin."