The claim followed Saturday's match between the teams at Anfield, and the Football Association have pledged to investigate.
Evra was reported to have told media in his native France that Suarez directed racist insults at him during the 1-1 draw.
Suarez wrote on his official Facebook page: "I'm upset by the acusations [sic] 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."