Evra, a black French defender, was quoted as telling French TV station Canal Plus: "There are cameras, you can see him (Suarez) say a certain word to me at least 10 times."
The FA confirmed it was investigating a claim of racism by Evra.
"Referee Andre Marriner was made aware of an allegation at the end of the fixture and has subsequently reported this to The FA," English football's governing body said. "The FA will now begin making enquiries into the matter."
Liverpool told The Associated Press: "The player categorically denies the allegations."
United did not immediately respond to a request for comment.
Javier Hernandez salvaged a 1-1 draw for United by canceling out Liverpool captain Steven Gerrard's free kick.