The 20-year-old defender broke down in tears during the League One outfit’s 5-1 loss to the Reds in Friday’s FA Cup tie after an apparent shout was directed at him from the stands.
A man from Aintree, also 20, has been arrested and released on bail in relation to the incident, and Liverpool have offered a full apology to Adeyemi and helped police.
Oldham insist they are delighted with Liverpool’s efforts and have praised their professional standards.
The statement from the club read: "The chairman and directors of Oldham Athletic Football Club would like to thank Liverpool Football Club and officers of the Merseyside Police for their concerns and painstaking efforts in investigating the circumstances surrounding the incident involving Tom Adeyemi.
"Excellent communication has been maintained and the club and player have been notified of every detail during the progress of the investigation. The professional standards applied throughout have been praiseworthy.
"The club would also like to thank the numerous fans from Liverpool and Oldham, and also those from around the country, who have sent letters of support to Tom.
"This type of incident is contained within a minority and should not deflect from a superb match that was enjoyed by both sets of club officials and fans. We wish Liverpool every success in the next round of the cup."