In a clip posted on Twitter, the South Korea international is asked by a man if he "does DVDs" - a reference that carries racial undertones.
Son attempts to laugh off the comment before driving away, but the actions of the person in question - who identifies himself as a Hammers fan - has been condemned by both clubs.
A Tottenham statement read: "Racist behaviour is completely unacceptable and we hope that the individual is identified to the authorities and the necessary action taken."
West Ham also hit out, with a spokesperson saying: "West Ham United have a zero-tolerance approach to any form of discrimination and unreservedly condemn the actions of the individual in this video.
"Anyone behaving in this manner is not representative of our club and its values, and anybody who witnesses discrimination or abuse such as this at West Ham are urged to contact us in confidence at firstname.lastname@example.org."
Son played the final 15 minutes of Tottenham's 2-0 defeat to Arsenal on Saturday afternoon.