Allardyce was reluctant to condemn supporters after Sunday's game at Tottenham as he had not heard the chants himself but, having seen the footage, he has backed his club's zero tolerance stance.
"I did not agree or disagree with what went on on Sunday, let me make that clear," he told Sky Sports News.
"I was doing a press conference after the game and I had no idea what had happened at the time because I had not heard or seen it as I was concentrating on the game.
"Since the game, it is obviously very disappointing to hear what has been said and done by a very small minority of fans.
"The good thing is with security cameras and CCTV cameras, it's difficult for them to get away with it. Hopefully we can pick out those people and punish them in the right way.
"Nobody can condone that sort of behaviour and there is zero tolerance of it at this club. The club, Tottenham and the authorities will deal with it accordingly."
Police arrested two fans during the match for making Nazi-style salutes, which officers described as a 'racially aggravated public order offence'.
One of those individuals has been identified by West Ham as a season ticket-holder and has been banned by the club.
"Any other individuals identified can expect a similar swift and robust response," a West Ham spokesman added.