Once the deal is complete, which many expect to happen this week, the fight will take place on March 31 at Principality Stadium in Cardiff, Wales.
"The possibility of this fight happening, I've visualised happening - fighting Anthony Joshua and knocking him out - and that's what I'm going to do".
"It is life-changing, not only for myself but for my parents and my family". Not many people realise what is involved with something like this, it's so big and complicated.
Joshua laughed alongside Norton and the studio audience, but while Parker has admitted to putting on weight recently following his victory over Hughie Fury in Manchester in September, he said he would quickly lose it again in his forthcoming camp. "There are so many people to say thank-you to". "It's not even in question for them", he told Sky Sports.
Parker left Auckland on Saturday night, a sure sign that his hugely anticipated world unification fight with the giant Brit will become a reality in the next couple of days.
"Listen, I'm not scared of the guy".
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"It's not signed yet but hopefully with this trip to the United Kingdom we get it signed", he says.
"I think if there are any fireworks it will be coming from him because obviously we did get under his skin and he got a bit angry". He should be able to accept the different challenges in boxing in terms of physical training, how people talk and what people say. I know that I can burn it off fast and get in shape fast. But styles make fights and I feel like his style matches me perfectly.
"A lot of people are underestimating me because of my last two or three performances. It will come down to the day, who is taking the hardest hits and who has the best day that day".
Hearn held another meeting with Parker's promoter David Higgins on Wednesday, as they continued to discuss the remaining details of a deal for Joshua to put his WBA "super" and IBF heavyweight titles at stake against the WBO champion.
Parker, who will be joined in London by trainer Kevin Barry, will fly to his Las Vegas base to prepare for the fight.