The allegations, first published in the Mirror, are reported to have happened between 1997 and 2000, police said, when Mills would have been in his mid-20s.
On Tuesday, the Metropolitan Police told BBC News that they had investigated Mills over allegations of serious sexual offences against a teenage boy who was under the age of 16.
In a statement on Wednesday afternoon, the 53-year-old said: “The recent announcement that I am no longer contracted to the BBC has led to the publication of rumour and speculation.
“In response to this the Metropolitan Police has made a statement, which I confirm relates to me.
“An allegation was made against me in 2016 of a historic sexual offence which was the subject of a police investigation in which I fully cooperated and responded to in 2018.
“As the police have stated, a file of evidence was submitted to the Crown Prosecution Service, which determined that the evidential threshold had not been met to bring charges.
“Since the investigation related to an allegation that dates back nearly 30 years and the police investigation was closed 7 years ago, I hope that the public and the media will understand and respect my wish not to make any further public comment on this matter.”
He added: “I wish to thank from the bottom of my heart all those who have reached out to me with kindness, my former colleagues, and my beloved listeners, who I greatly miss.”
In his statement, Mills does not address the substance of the allegations, provide any further detail about them or offer a denial.
He also does not go into what happened at the BBC and the reasons for his sacking.