"Whilst the incident occurred in the home end, we are yet to determine if the alleged individual was a Tottenham Hotspur supporter or not," a Spurs spokesman said.
"Any kind of racist, discriminatory or anti-social behaviour will not be tolerated by the club.
"We have a strict, zero tolerance attitude in this regard and shall take action against anyone behaving or using language that is abusive, offensive or obscene."
Dickens said he is working with the club to identify the supporter.
"I've not experienced racism at football for about 15 years," Dickens wrote on Twitter. "But today at @SpursOfficial was hideous. Reported to stewards, confronted him myself.
"Spurs have been in touch and I'll work with them to find the person," he wrote. "Stand up to racism and it won't survive, let it go and it will just get worse."
Last month, Spurs manager Mauricio Pochettino said he wanted to "kill" bad behaviour and racism from the stands after a fan threw a banana skin on to the pitch during the 4-2 defeat at Arsenal.
Three Chelsea supporters were arrested for racially aggravated public order offences at the League Cup semi-final first leg between Tottenham and Chelsea last Tuesday.
Spurs' 1-0 defeat to United was Son's last game with the club this month as he is joining South Korea for their Asian Cup campaign in the United Arab Emirates.
(Reporting by Hardik Vyas in Bengaluru; editing by Clare Fallon and Toby Davis)
