The unknown presenter is reportedly a BBC star who has been taken off air since the allegation emerged.
The mother of the alleged victim said the money - allegedly amounting to more than £35,000 - was used to fund a cocaine addiction which “destroyed” her child’s life.
After the story was published, Mr Clark said: “Not sure why my names floating about but re that story in the sun- that ain’t me babe. I’m Currently filming a show in Italy for the bbc, so take my name out ya mouths.”
Radio star Jeremy Vine also responded to the story, telling his Twitter followers: “It certainly ain’t me”.
Addressing more baseless speculation, he wrote: “Just to say I’m very much looking forward to hosting my radio show on Monday — whoever the “BBC Presenter” in the news is, I have the same message for you as Rylan did earlier: it certainly ain’t me.”
The family of the alleged victim complained to the BBC about the behaviour on 19 May and begged the broadcaster service to make the presenter “stop sending the cash,” according to The Sun.
Neither the individual or the teenager, who was said to be 17 when the payments began, was identified.
The BBC said it takes “any allegations very seriously”.
Sympathetic Twitter users responded to Mr Clark’s tweet, with one person saying: “Ignore them. This kind of reporting isn’t fair on the majority of decent presenters. Keep doing you Rylan x”
Another said: “You have always come across as 100% genuine. Don’t worry about it.”
In response to The Sun’s report, a BBC spokesperson said: “We treat any allegations very seriously and we have processes in place to proactively deal with them.
“As part of that, if we receive information that requires further investigation or examination we will take steps to do this.
“That includes actively attempting to speak to those who have contacted us in order to seek further detail and understanding of the situation.
“If we get no reply to our attempts or receive no further contact that can limit our ability to progress things but it does not mean our enquiries stop.
“If, at any point, new information comes to light or is provided – including via newspapers – this will be acted upon appropriately, in line with internal processes.”