The 18-year-old midfielder returned from under-19 international duty with a hamstring injury after playing three matches of 90 minutes in five days.
Klopp said the situation was not normal and he would be speaking to someone at the Football Association.
“Rossiter is a special story. I never heard of an 18-year-old playing three games in five days,” he said.
“That is the problem why he is injured and I don’t think he will be ready until the next international break.
“I don’t know who I have to talk to about this but I will find a way because it is not okay. On my first day I didn’t want to have a call with someone with the FA but for sure this is not okay.
“These young players are our future. If we handle them like horses we get horses.”
Klopp is likely to make changes to his side for tonight’s Europa League visit of Russians Rubin Kazan as he is holding back the likes of striker Daniel Sturridge, who sat out the weekend draw at Tottenham with a knee problem, for Sunday’s visit of Southampton.
Klopp’s home debut is one of the most eagerly-anticipated in the club’s history but the German sought to manage expectations.
“We have to be patient, in a football way. I hope no-one is waiting for it to be 4-0 after three minutes and is then disappointed,” he said.
Meanwhile, Tottenham head coach Mauricio Pochettino has a decision to make over whether to field Harry Kane in their Europa League clash against Anderlecht this evening.
The England international has travelled with the squad to Belgium but Pochettino has said he suffered a knock against Liverpool in Saturday’s draw at White Hart Lane.
Son Heung-min (foot), Federico Fazio (hamstring), Nabil Bentaleb and Nacer Chadli (both ankle) are all missing for Spurs but Eric Dier is available having sat out Saturday’s game through suspension and Ryan Mason took part in training as he continues his recovery from a knee injury.