Bristol City owner Steve Lansdown says the club will ban any fan found to have used racist language during Saturday's 3-0 Championship defeat at Luton.
Lansdown said he was "shocked and disappointed" to learn of reports of abuse coming from the away end.
An FA Cup tie between Haringey Borough and Yeovil was abandoned on Saturday amid reports of racial abuse by fans, which is being investigated by police.
Lansdown said the Robins were a "family club which celebrates diversity".
In a statement he added: "I can assure our genuine supporters and Luton Town, who have been very helpful, that we are fully investigating these reports.
"We will deal harshly with anyone found behaving in a racist manner and they will be banned from Ashton Gate and from purchasing tickets at future away matches.
"Racial intolerance is not something that we want being associated with our club in any shape or form."
Luton released a statement on Sunday saying they would "support any investigation taking place between Bristol City and the authorities".
It continued: "Hatters CEO (chief executive officer) Gary Sweet has already been in contact with his counterpart at Ashton Gate, Mark Ashton, to express our support and co-operation.
"Like the Robins, Luton Town are a family club that celebrates diversity and inclusion, and will not accept any discriminatory comments or behaviour at our matches."