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 https://www.bbc.co.uk/sport/football/43860354">PFA Player of the Year, https://www.bbc.co.uk/sport/football/43960328">Football Writers' Footballer of the Year and https://www.bbc.co.uk/sport/football/44099913">Premier League Player of the Season last year after his 32 goals set a new record for the most in a 38-game season.
He has been praised by Steve Rotheram, mayor of Liverpool, for https://www.bbc.co.uk/sport/football/44585980">helping to tackle Islamophobia and "breaking down barriers", as well as being https://www.bbc.co.uk/news/world-43921603">described as a role model for Muslim children.
