Egypt striker Salah, 26, was filmed on a mobile phone from a section of home supporters as he was taking a corner, BBC Sport reported.
The footage, taken by a fellow fan, shows expletives directed at Salah including about his Muslim religion.
In a statement, West Ham said it has "a zero tolerance policy to any form of violent or abusive behavior".
It continued, "We are an inclusive football club.
"Regardless of age, race, religion or belief, marriage or civil partnership, pregnancy or maternity, sex, sexual orientation, gender reassignment or disability, everyone is warmly welcomed at London Stadium.
"Anyone identified committing an offence will have their details passed to the police and will face a lifetime ban from London Stadium.
"There is no place for this kind of behavior at our stadium."
The user who posted the video on Twitter said, "I was disgusted by what I was hearing. People like this deserve no place in our society let alone football matches."
Police are aware of the incident.