Duco spokesman Craig Stanaway said Price was the "favoured option because Parker is keen to fight in the UK."
Price told Sky Sports: "My management contacted me...and said would I be interested in fighting Joseph Parker.
Price, who won an Olympic bronze in the 2008 Beijing Olympic superheavyweight class, has won 21 of his 24 professional fights, 18 by knockouts.
Parker and Price were due to fight on the Anthony Joshua this year undercard but the clash fell through.
Price said: "I've already agreed to fight them before and now he's world champion, it makes it even more of an attractive proposition to fight him. Now I'm just waiting for news really."
Price was apparently less than impressed by Parker's world title victory over Mexican-American Andy Ruiz jnr.
"There must have been a lot of nervous tension fighting for a world title, because he's looked better in his previous fights," said Price.
"You only have to look at his fight against [Alexander] Dimitrenko to see him deal with a taller man, but I do believe I'm a different proposition.
"I've got plenty to show which hasn't been shown. I've been written off by a hell of a lot of people.
"You've got to be realistic and they will be using me as a chance to show themselves to the British public, and they obviously see me as somebody they can look good against.
"But I don't take offence from that. I'll just be made up to have the opportunity to achieve a career-long dream and become a world champion."
One boxing publication has already doubted Price's credentials while also dissing the New Zealander's performance against Ruiz.
Writing for boxingnews24, Jeff Aranow said: "Price has had problems with his punch resistance during his career, and there would be a good possibility that he would fold quickly under the big power shots from Parker if he chose to fight aggressively.
"That's a big if though, because Parker was not aggressive in his close and some would say controversial 12 round majority decision win over the Top Rank promoted Andy Ruiz Jr.
"Parker was very passive in the fight, and outworked and out-landed in virtually every round of the fight. Parker appeared to lose the vast majority of the rounds in the eyes of a lot of boxing fans."
But Aranow believed Price had the punching power to worry Parker early on if the fight did go ahead.Veteran promoter Frank Warren, who is handling Hughie Fury, told the Daily Telegraph:
"We want the Parker-Fury fight in the UK, they want the fight in New Zealand, but we are still talking with Parker's team. The UK is the place to have heavyweight fights at the moment.
"With the WBO making (David) Haye their No 1 challenger it becomes very interesting. He would have no fears about facing the winner of Fury-Parker. And with Tyson Fury hopefully coming back next year, if Haye becomes a world champion, that is another huge fight to be made."
