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 appeared as West Ham substitutes in Saturday’s 4-2 home defeat by Manchester United.
Wets Ham are reviewing CCTV footage and speaking to witnesses.
The club 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.
“I believe all human beings are equal. We are all born with different physical attributes.
“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.