A Chelsea fan was arrested during Sunday's Premier League game at Tottenham for allegedly racially abusing Son Heung-Min.
It was the only arrest for racism made by the Metropolitan Police at the London derby where Chelsea defender Antonio Rudiger reported being subjected to monkey chants.
Rudiger was involved in an incident which saw Son sent off after a VAR review in Chelsea's 2-0 victory. The Germany international indicated afterwards he had been the victim of racist abuse from a section of the home support.
There was no immediate indication when any racism toward Son happened.
"Chelsea Football Club will not tolerate any form of racism, nor is there a place at this club for anyone who is found to have engaged in such behavior," a statement said.
"If a season ticket holder or member is involved, we will take the strongest possible action against them including bans. We will also support any potential criminal proceedings by the relevant authorities."
Tottenham said on Monday that their investigation into the racism aimed at Rudiger had been "inconclusive" as it worked to look into the abuse of Son.
A club statement read: "We are able to track every fan via the cameras and have spent many hours reviewing CCTV footage. We have engaged lip readers to study the footage and contacted Chelsea for further information from their players.
"We have also taken statements from other parties present at the time. The police will be reviewing our evidence alongside us. Please be assured we shall be exhaustively investigating this matter.
"This club has a proud track record of anti-racism work across all our communities and we are determined to ensure that we conduct a thorough investigation. Any fan found to be guilty will receive a lifetime ban. At this time however we should point out that our findings are inconclusive and would ask that comment is reserved until the facts are established."
Meanwhile, Spurs have issued an immediate ban to a supporter who threw a cup towards Chelsea goalkeeper Kepa Arrizabalaga during the match.
A statement read: "We shall continue to take the strongest action possible against anyone found to be behaving in this way."