Finlayson has apologised for the incident, which happened in the third term of Port's 69-point win over the Bombers on Friday night at Adelaide Oval.
It is understood an umpire's microphone picked up the insult and at least one Essendon player also called it out.
Port said in a statement that Finlayson will make a statement on Sunday, with the target of the slur yet to be identified.
"A contrite Finlayson made the club aware during the three-quarter-time break and apologised to the victim on field after the final siren last night," the Power said.
"Finlayson, who will be counselled by club leaders in the coming days, will provide a statement tomorrow.
"The club will now wait for the AFL to complete its investigation."
The AFL also confirmed on Saturday night that Finlayson is under investigation.
"We are taking this matter extremely seriously," an AFL spokesman said.
"AFLIU (integrity unit) are now investigating and we will provide an update once that investigation is completed."
Last month, North Melbourne coach Alastair Clarkson was fined $20,000 and has a suspended two-match ban for abusing St Kilda defenders Jimmy Webster and Dougal Howard during a pre-season game.
Clarkson was asked to explain his actions to the AFL after allegedly using the homophobic slur "c*******er" during the practice match.
20°C