Egypt striker Salah, 26, was filmed on a mobile phone from a section of home supporters as he was taking a corner.
The footage, taken by a fellow fan, shows expletives directed at Salah including about his Muslim religion.
In a statement, West Ham said they have “a zero tolerance policy to any form of violent or abusive behaviour”.
It continued: “We are an inclusive football club. 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 behaviour 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.
“We are dismayed to see yet another high-profile incident of discrimination in English football,” anti-discrimination charity Kick It Out said in a statement.
“We are pleased that West Ham officials have expressed their determination to identify the supporter and take firm action – this type of behaviour is simply unacceptable and must be challenged swiftly and decisively.”
Salah was named PFA Player of the Year, Football Writers’ Footballer of the Year and Premier League Player of the Season last year after his 32 goals set a new record for the most in a 38-game season.