The statements were made during NZME Radio's KICK Fresh Music Friday podcast on March 25.
A joint statement from Gayford and NZME said those rumours - also published on KICK's social media page - were "damaging and untrue".
"The statements were based on rumours about Mr Gayford that are baseless lies," the statement published this afternoon on KICK's social media page said.
"NZME Radio has apologised to Mr Gayford for these publications and the hurt and distress they have caused and accepts that he has never been the subject of criminal charges and is not now the subject of criminal charges in any court in New Zealand."
"In a settlement between the parties, NZME Radio agreed to pay a confidential sum to Mr Gayford."
Gayford said he would be "making no further comment."
A spokeswoman for NZME said the company would make also not comment beyond the statement.
In 2018, Police Commissioner Mike Bush took the extraordinary step of signing off a media release saying Gayford "is not and has not been the subject of any police inquiry, nor has he been charged in relation to any matter".
Police do not usually respond to enquiries seeking to confirm if individuals are under police investigation, the statement said at the time.
The move came after the fishing show host had been under an unprecedented assault of baseless rumour and false innuendo with the apparent intent of dragging down the Prime Minister.
For seven months, Gayford had been the subject - on social media and via word of mouth - of untrue allegations and accusations.
KICK is owned by NZME, which also publishes the New Zealand Herald.
