Play was stopped during Chelsea’s win at Tottenham after German defender Rudiger said he heard monkey noises.
A Spurs statement said the club and the Metropolitan Police had “now exhausted all avenues of investigation”.
Spurs added that the police had closed the crime report on the matter.
The Tottenham statement added: “We fully support Antonio Rudiger with the action that he took – however there is no evidence to corroborate or contradict the allegation and as such neither ourselves nor the police are in a position to take any further action.”
A total of six arrests were made following the match at the Tottenham Hotspur Stadium as part of the Metropolitan Police operation at the fixture, but none were linked to the incident involving Rudiger.
One Chelsea fan was arrested for a racially aggravated public order offence against Spurs forward Son Heung-min, who was sent off in the match after a collision with Rudiger.
Spurs said they are 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”.
On Monday evening, Chelsea issued a statement which began: “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.”
Earlier Tottenham explained that they had worked with professional lip readers in their investigation and that all reports had also been reviewed by the police.
Their statement added: “We are fiercely proud of our anti-racism work and our zero tolerance of any form of discrimination. 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 fan base.
“If any new information comes to light, this will be fully investigated.”