Evra said after the game that Suarez used the same insult "more than 10 times" throughout the highly-charged clash on Merseyside.
Suarez released a statement saying: "I'm upset by the accusations of racism. I can only say that I have always respected and respect everybody.
"We are all the same. I go to the field with the maximum illusion of a little child who enjoys what he does, not to create conflicts."
The Uruguayan faces an FA investigation after referee Andre Marriner included Evra's claims in his match report.
Evra told Canal Plus: "I was very upset. In 2011 you can’t say things like this. He knows what he said, the ref knows it, it will come out.
"I won’t repeat what he said, but it was a racist word, and he said it more than 10 times. He tried to wind me up. I won’t make a huge deal out of it, but it’s very upsetting and disappointing."
The pair clashed midway through the second half when Suarez caught Evra on the knee and the latter was later booked for dissent.
The match ended in stalemate after Steven Gerrard's opener was cancelled out by United substitute Javier Hernandez, with the result leaving the champions in second place behind new league leaders Manchester City, who beat Aston Villa later in the day.
A Liverpool spokesman said: "The first we knew about these allegations was 20 minutes after the final whistle when the manager was asked to go into the referee’s office and told about them.
"The first thing we did, as you would expect, is ask the player and he has categorically denied using any language of that nature."
The FA confirmed they would be looking into the incident.
A spokesman said: "The FA has been made aware this evening of an incident that is alleged to have occurred during the Liverpool versus Manchester United fixture at Anfield.
"Referee Andre Marriner was made aware of the allegation at the end of the fixture and has subsequently reported this to the FA. The FA will now be making enquiries into the matter."