The on-loan Norwich City defender was left close to tears over the incident last week.
Oldham thanked both parties for their continued communication and support throughout the investigation to date.
"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," the statement read.
"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 statement continued: "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."