The Reds beat Arsenal 5-4 on penalties in a thrilling fourth-round tie on Wednesday evening that ended 5-5 after 90 minutes.
The quarter-final ties are due to take place on the week commencing December 16, but the European champions are due to be heading out to Qatar for the Club World Cup that week.
And Klopp insists they could forfeit their tie unless an alternative date is found for them to play.
“If they don't find an appropriate place for us, not 3am on Christmas Day, then we don't play it," Klopp remarked.
“They have to make that decision. You have to think about these things.
“If you have a fixture list where one team cannot be part of all the games, then you have to think about the fixture list.
“Hopefully it starts now. I really think that's fair, and this problem is obvious now. We will not be the victim of this problem.
“We play tonight, we wanted to win it, we did that, and if they don't find a proper date for us, then we cannot play the next round and whoever is our opponent will go through. Or Arsenal play them.”
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