Klopp surprisingly left Mane out of the starting line-up for the clash against the Premier League champions, with the Senegal international only being introduced off the bench in place of Alex Oxlade-Chamberlain towards the closing stages at Anfield.
Mohamed Salah opened the scoring against his former club midway through the second half at Anfield, before Willian equalised in the 85th minute to give the visitors a share of the spoils.
Mane was caught by television cameras having an argument with Klopp at the end of the match, but the Senegal forward insisted the exchange was normal and nothing serious.
"[Klopp] asked me to play on the right side and one time I tried to ask for the ball in the number nine position," Mane explained to Sky Sports.
"Mo (Salah) was on my side and the coach was shouting 'Sadio come onto the right side'.
"After the game I told the coach it was easier if you told Mo to come in because he was close to you.
"But it was friendly, it was not because I was unhappy or something else. There was no problem, I was just communicating and explaining that to him after the game."
Mane went on to praise the impact made by Salah, who has scored 10 goals in 13 league appearances for the Merseyside club after arriving on a £34m ($45m) transfer from Roma in the summer.
"He's doing very well for us and we are happy to have him in the squad," Mane said. "He is a very important player for us and I'm going to keep trying to help him score more goals for the team.
"He is a very, very good player for us and [if you give him] space he is unbelievable. No one can catch him and also he is clever."