In the closing stages of Liverpool's 5-1 FA Cup third-round victory last Friday night, Adeyemi - on loan at Oldham from Norwich - appeared to take offence to something shouted from the Kop and alerted referee Neil Swarbrick before being comforted by players from both sides.
On Sunday, police investigating the incident arrested a 20-year-old man from Aintree, Merseyside on suspicion of a racially aggravated public order offence. He was later released on bail.
Liverpool also issued an official apology to Adeyemi and, in a statement on their website, 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."