Bristol City are investigating reports of "racist language used in the away end" during their 3-0 Sky Bet Championship defeat to Luton Town at Kenilworth Road.
It comes following the abandonment of an FA Cup qualifying game between Haringey Borough and Yeovil Town after players walked off the pitch due to racist abuse.
Luton ended City's 10-game unbeaten run on Saturday, as Lee Johnson's side slipped to ninth in the table.
In a statement, the Robins said: "Bristol City Football Club has been made aware of allegations of racist language used in the away end at Saturday's Sky Bet Championship fixture at Luton Town's Kenilworth Road (October 19th).
"The club naturally condemns any form of abuse or racist language. Bristol City are a family club that celebrates its diversity and inclusivity.
"Action will be taken against anyone behaving in a racist manner at a Bristol City match, who has purchased their ticket or season card through the club as well as reporting them to the relevant authorities.
"The club are now fully investigating these reports and can assure supporters that appropriate action will be taken."
Anti-racism organisation Kick It Out tweeted in support of City's stance, while the Robins' owner Steve Lansdown also re-affirmed the club's condemnation of the alleged incident at Kenilworth Road.
"I am shocked and disappointed to hear of these allegations," he said.
"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.
"I firmly believe that, in no way, is it representative of our true supporters or the essence of Bristol City as a family club which celebrates diversity and inclusivity."
We fully support @BristolCity's uncompromising stance on racist abuse, and praise their committment to fully investigating the incident.— Kick It Out (@kickitout) October 20, 2019
Those responsible should face strong punishment and receive education to learn from such behaviour. #KickItOut https://t.co/dv2TkU6ctA
Luton have also confirmed they are aware of the allegations: "Further to national media reports of alleged racist language used in the away section of Kenilworth Road yesterday, Luton Town Football Club would like to make clear that we will support any investigation taking place between Bristol City and the authorities.
"Bristol City have released a club statement about the allegations and Hatters CEO 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."