The Reds' games in the Bundesliga Cup youth tournament against Hoffenheim on Friday and Eintracht Frankfurt on Saturday were abandoned as a result of the alleged racism towards the same player.
A club statement said: "Liverpool can confirm that a member of our under-18s team reported he was racially abused by an opponent while playing in the Bundesliga Cup youth tournament at Schwabisch Hall in Germany on Friday.
"The player immediately alerted the match official, his teammates, and our coaching staff. Consequently, LFC's management team decided to stop the game and leave the pitch together.
"In a further incident, the same player was the target of abuse during a fixture in the tournament (on Saturday). Again, the player alerted his team-mates and coaching staff and our management team opted to leave the pitch once more."
Liverpool backed their players' decision to leave the pitch on both occasions and called for an investigation into the incidents.
"We are proud of our player for his prompt actions in reporting the incident and the maturity of his response. He and any of his team-mates affected are being supported by the academy safeguarding team," the club said.
"The club calls upon our opponents and the tournament organisers to conduct an urgent and thorough investigation into the incident."
Speaking to German newspaper Bild, Eintracht sporting director Timmo Hardung said the club took the accusation "seriously" but insisted it had been a "linguistic issue" relating to a term that is commonly used in Germany.
"Our entire youth academy is like Eintracht Frankfurt, multicultural and we have clear values that we live by and demand again and again," Hardung said.
"Racism has no place here, but we still take such an accusation seriously.
"We can rule out racism, it is a linguistic issue and therefore hope to have resolved the misunderstanding among the players and Liverpool officials concerned."