Evra is reported to have told media in his native France that Suarez directed racist insults at him during the 1-1 draw at Anfield.
The FA then confirmed that they are examining the claims after referee Andre Marriner was also made aware of them.
A statement from the FA said: "The Football Association have this evening been made aware of an incident that is alleged to have occurred during the Liverpool versus Manchester United fixture at Anfield today.
"Referee Andre Marriner was made aware of an allegation at the end of the fixture and subsequently reported this to the FA.
"The FA will now begin making enquiries into the matter."