The 19-year-old replaced Daniel Sturridge at half-time at Prenton Park, in one of 10 planned changes by Klopp as the Reds began their pre-season campaign.
While the former Chelsea striker could not beat Tranmere goalkeeper Scott Davies, he produced some strong moments in the final third.
Most notably this came with his parried effort that resulted in a goal for Pedro Chirivella, and a fine touch and strike on the turn that drew Davies into a fine stop.
It is still up in the air whether Solanke will turn out for the first team or the U23s this season, but speaking after the game, Klopp suggested he could force his way into contention.
“He decides that himself. He is a young lad and that means it’s a long-term project,” he said.
“I have all the time in the world, but if he brings a little bit of a rush with his development, then everything is good.
“I saw a few wonderful signs tonight. Everybody in England watched the U20 World Cup—that was fantastic football.
“He is an offensive player, a really skilled boy. He has to improve, of course, but a lot of things are already really good: first touch, movement, a lot of things.
“So it’s about him, if you want, and the situation. But I will not avoid his development, so the moment he is strong enough he is there.
“Where he is training isn’t already decided because I didn’t decide for all of the team.
“But he will train with us as often as makes sense and is possible. And where he will play, we will see.”
Addressing the Reds’ relative inactivity in the market since the arrivals of Solanke and Mohamed Salah, Klopp hinted that things were moving behind the scenes.
“We cannot only buy players because other teams buy players—we do our business as well as we can do it and we are completely in it,” he added.
“We are convinced about the way we are going, so that’s all. I’m not nervous, maybe that’s the right message.”
Liverpool‘s next pre-season clash comes away to Wigan Athletic on Friday night, and Solanke will be hoping to start at the DW Stadium after his impressive cameo in Birkenhead.