Tottenham say no evidence was found to support allegations of racism from their fans towards Chelsea defender Antonio Rudiger during the sides' Premier League match last month.
Play was stopped during the game after Rudiger said he heard monkey noises.
Spurs say they "fully support Antonio Rudiger with the action he took" and the club and police "exhausted all avenues of investigation".
"There is no evidence to corroborate or contradict the allegation," they said.
Chelsea said: "We support Toni Rudiger totally and unequivocally on this matter, and as Tottenham's statement makes clear, a lack of evidence does not mean an incident did not take place.
"In responding to this incident, we must be very careful about the climate we create for players who experience and report racist behaviour.
"It is vitally important that we continue to encourage all players, whatever shirt they wear, to report racist abuse without fear of doubt or reprisal."
A total of six arrests were made following Chelsea's win at the Tottenham Hotspur Stadium on 22 December as part of the Metropolitan Police operation at the fixture, but none were linked to the incident involving Germany international Rudiger.
One Chelsea fan was arrested for a racially aggravated public order offence against Spurs forward Son Heung-min, who had been sent off after a collision with Rudiger.
Spurs said they were able to "track every fan" using cameras at their new 62,062 stadium and that any supporter found to be guilty of racism would "receive a lifetime ban".
In their statement on Monday, the club said they had worked with professional lip-readers, and that all reports had also been reviewed by the police.
"We are fiercely proud of our anti-racism work and our zero tolerance of any form of discrimination," the club said. "This is one reason why we have attributed so much time and resource to investigating this matter.
"Had we identified anyone guilty of this we were intent on issuing them with a lifetime ban from our stadium as they would have no place among our proud, diverse fanbase.
"If any new information comes to light, this will be fully investigated."
They said the police had notified them that "they have closed the crime report as they can find no evidence to support the allegation of racial abuse".