Liverpool secured their sixth European Cup with a win over Tottenham in the final in Madrid in June.
But they opted not to parade the trophy on Friday, when they opened their Premier League campaign with a 4-1 victory over Norwich at Anfield.
Asked about that decision, Klopp was seemingly unaware of the possibility, but said he would not have done it anyway.
"Nobody told me I have to. Is that what you do usually? I touched the Euro cup on the matchday there, the day after, and since then not," he told a news conference.
"No reason for it, I know how it feels. And people know how it looks.
"If somebody expected that then somebody should've told me. I'm pretty sure I wouldn't have done it anyway, because we are now in the new season. Enough celebration.
"Nobody has to congratulate me anymore, but still everybody's doing it. I'm really fine with all the things I heard about it, now let's play the new season."
A calf injury to goalkeeper Alisson soured Liverpool's win , which came thanks to first-half goals from Mohamed Salah, Virgil van Dijk and Divock Origi after Grant Hanley's seventh-minute own goal.
But Klopp says Adrian, who came in for the injured Alisson, is ready to play.
“He’s a confident guy,” the Reds boss said. “I don’t think he wanted to sit on the bench for the next couple of years, he made himself ready.
“It’s a situation, but if you’re in a situation like this you want to have somebody who’s played in the league, who’s experienced.
“He’s 32, a calm guy, good with his feet as we saw. That’s important for us, so a lot of good things, but you still don’t want to have a situation where only one goalie is fit.”