The Scotland international allegedly told a man to "f*** off back to your own country ya clown" during an angry online exchange.
The 22-year-old, who is on loan at Hibs from npower Championship side Wolves, posted the comment to almost 13,000 followers late on Thursday night.
It has subsequently been removed from his Twitter page.
Now Lothian and Borders Police have confirmed they have carried out early enquiries after they were made aware of Griffiths' alleged comments.
A spokesman said: "We have been made aware of this matter, and we are carrying out initial enquiries."
Griffiths was responding to a tweet from a Twitter user believed to be based in Livingston, West Lothian.
The user had earlier sent a tweet to Griffiths. It said: "@LeighG29 @jordan-cosgrove thumb boy why do you make fun of other peoples looks when you are the worst out the lot?"
Griffiths later insisted he was not a racist, posting: "Got to apologise to @zak-iqbal for my comment earlier, was out of order!"
This morning, he also wrote: "I'm not a racist! Yes what I said was out of order! People going to give it a rest now?? Jesus Christ."
Griffiths is currently on loan at Easter Road but is due to return to Molineux following the Edinburgh club's game with Dundee on January 19.
He has scored 14 goals for Hibs this season and boss Pat Fenlon is keen to keep him for the remainder of the season.
However, a club spokesman today confirmed that Hibs would be taking action against the former Livingston and Dundee forward.
He said: "Hibernian FC today confirmed that it had spoken to player Leigh Griffiths about exchanges on a social media website.
"The player had realised that despite the provocation, the language he used was totally unacceptable and had already apologised for what he said. The player has closed access to the account.
"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."
A Wolves spokesman added: "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."
Griffiths' Wolves deal expires at the end of the season but the club have an option to extend it until the summer of 2014.
Wolves paid Dundee £150,000 to sign Griffiths in January 2011 but his only appearance for the club came in the League Cup tie against Northampton the following August before he was farmed out to Hibs for his first loan spell at Easter Road last season.
PFA Scotland released the following statement: "Despite the fact that players are often subjected to unacceptable levels of personal abuse on social media sites they should not retaliate to such abuse.
"PFA Scotland therefore strongly condemns racism and all discriminatory behaviour in whatever form that takes.
"PFA Scotland is a founder member, long time supporter and sponsor of Show Racism the Red Card and remains hugely supportive of the educational messages they provide."