The report came after TVNZ confirmed Hilary Barry and Jack Tame would replace Rawdon Christie and Nadine Chalmers-Ross next month in a bid to boost Breakfast's ratings battle against Paul Henry.
Henry addressed the report on Monday morning, denying any changes were taking place.
"It's not true at all. I have on many occasions been offered 7 o'clock, not just by this network but by the opposition network, TVNZ, who practically begged me to do it," he said. "On every occasion I said no, because I don't want to do it."
Read more: Paul Henry says TVNZ 'begged' him to host a 7pm show
Garner and du Plessis-Allan also addressed the rumours on Story from behind those masks.
"This is the new-look Story. Really hope you like it," said du Plessis-Allan.
"Facts used to kind of matter in reporting, didn't it?" continued Garner. "You're stuck with us for a little bit longer yet, well we think so anyway."
* Watch last night's episode of Story here.