In a statement on the club's website, Burnley said: "During the game, one male Burnley supporter was asked to leave the stadium following repeated racial comments aimed at a Bradford player, who reported the incidents at the time to the officials.
"The person involved denied any racist behaviour, but witnesses have confirmed racial comments were made. As the supporter was ejected, another male supporter left voluntarily at the same time. No arrests were made.
"Burnley Football Club would like to make it clear that we have a zero tolerance approach to any such behaviour and strongly condemn discrimination of any type. We would like to offer our sincere apologies to the player involved and to Bradford City Football Club."
The club added that it will now work with police, who have CCTV footage of the incident, to identify the fans in question.