The midfielder, who is on loan to the Latics from Norwich City, was allegedly the target of abuse from a Liverpool supporter during the side's 5-1 defeat to the Reds, which prompted a police investigation.
The Merseyside club apologised to the 20-year-old and stated that they were committed to punishing the supporter in question if found guilty of an offence.
Oldham released a statement on the club's official website, which said: "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."
A 20-year-old man was arrested by police over the incident and was later released on bail.