Juve have dominated Serie A with eight straight title wins, but their last success in Europe's premier club competition came in 1996.
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 LaLiga 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!"