The comments appeared on an unverified Twitter account bearing Griffiths' name, which has over 12,000 followers. Griffiths is currently on loan at Scottish Premier League side Hibernian from the Championship club.
The comment attributed to Griffiths, sent to user @Zak_Iqbal, read: "f*** off back to your own country ya clown".
Wolves said in a statement: "Wolves are aware of an alleged racist comment made on Twitter last night by Leigh Griffiths.
"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."