Five Hammers fans were arrested at White Hart Lane and a further two were cautioned for displaying Nazi salutes, one of which was a season-ticket holder that West Ham have now banned for life.
A West Ham statement read: "Five supporters were arrested outside of the stadium for unrelated incidents and two more were cautioned for racially aggravated gesturing inside the ground.
"One of those fans has been identified as a season ticket holder and has been sent a letter containing a banning order from the club. Any other individuals identified can expect a similar swift and robust response."
The away supporters also taunted the home side over the attacks on Spurs fans ahead of their Europa League clash at Lazio on Thursday, in which one man was stabbed.
An FA statement read: "The FA governance department contacted both clubs and will continue to make enquiries into the matter.
"We note the statement issued by West Ham and encourage clubs to identify and ban for life any individuals involved in incidents of abusive chanting.
"There is no place for anti-Semitism or any form of discrimination in football. The FA is committed to playing our part in addressing this unacceptable behaviour."
The FA will liaise with the police during their investigations, and West Ham and Tottenham will also join forces as they compile reports into the unsavoury scenes.
A Spurs club spokesman said: "We are currently compiling a full report for the Football Association and shall be submitting this with all our evidence including relevant CCTV footage.
An earlier statement issued by West Ham read: "We are in contact with Tottenham to assist them with their investigation into the conduct of a small number of supporters and alleged inappropriate chanting during the match.
"We will take the strongest possible action against any of the supporters, including enforcing life bans from the club, if found guilty of behaviour which is categorically not condoned by West Ham United."
Peter Herbert, the chairman of the Society of Black Lawyers, had earlier confirmed that an official complaint had been lodged with the police, while anti-racism group Kick It Out are also looking into the affair.