Griffiths, currently on loan from Championship side Wolves, responded to a provocative tweet by swearing at user Zak Iqbal and telling him to go "back to your own country".
Lothian and Borders Police have confirmed they have made initial inquiries regarding the incident, but Hibs have revealed the player will be disciplined internally for his actions.
A club statement read: "The player supports the club and the whole of Scottish football in its stance against racism, bigotry and anti social behaviour.
"The player will be subject to a disciplinary process which will be a private matter between the club and the player."Wolves have also expressed their disappointment with the 22-year-old, who later apologised for the tweet and admitted his words were "out of order".
"Wolves are aware of an alleged racist comment made on Twitter last night by Leigh Griffiths," a club spokesman said.
"The club condemns such comments in the strongest possible terms and are extremely angry and disappointed.
"The player is currently on loan at Hibernian FC and is likely to remain there for the rest of the season.
"He is, therefore, subject to the disciplinary procedures of Hibernian FC and those of the Scottish Football Association.
"However, Wolves are in dialogue with Hibernian and are making their feelings known on the matter."Griffiths has trained at Molineux this week, but will likely remain north of the border where he is Scottish football's joint top scorer at the halfway point, scoring 14 goals in all competitions.