The incident took place on a flight to Belgium, where West Ham were playing Genk in the Europa League.
West Ham drew the game 2-2 on Thursday and have qualified for the knockout stage of the competition.
“West Ham are appalled by the contents of the video circulating on social media and condemn the behaviour of the individuals involved,” the club said.
They added they are “liaising with the airline and relevant authorities to identify the individuals”.
The club continued: “We continue to be unequivocal in our stance – we have a zero-tolerance approach to any form of discrimination.
“Any individuals identified will be issued with an indefinite ban from the club.
“Equality, diversity and inclusion are at the heart of the football club and we do not welcome any individuals who do not share those values.”