West Ham are investigating a report that the families of Victor Obinna and Frederic Piquionne were racially abused by fans at Upton Park on Saturday.
The pair, originally from Nigeria and New Caledonia respectively, appeared as West Ham substitutes in the 4-2 defeat by Manchester United.
West Ham are reviewing CCTV footage and speaking to witnesses.
They said in a statement that anyone found guilty of racial abuse "will be dealt with appropriately".
Co-chairman David Sullivan told the Evening Standard he would be "appalled" if the allegations of racist abuse proved correct.
"It is particularly sad that the families of our players were so treated," he said.
West Ham's supporter plan states that "there is no place for racism, prejudice and abuse in football" and it warns that in the past fans have been banned from the Boleyn Ground and convicted on criminal charges for their behaviour at matches.
"I believe all human beings are equal. We are all born with different physical attributes," added Sullivan.
"We will be monitoring CCTV, interviewing stewards and taking strong action against offenders if these allegations are proved to be correct."
It has been reported that a relative of Obinna, who had been watching the match from an executive box, confronted spectators about the abuse.
West Ham are one of 30 professional clubs to have achieved the first level of the Racial Equality Standard set up by the Kick It Out campaign.