Negotiations between Parker's promoters Duco Events/Top Rank and Fury's Frank Warren will enter a purse-bid process next week, with the highest bidder winning all rights to the contest.
If Parker's camp wins, the bout will likely take place in New Zealand in early April, while Warren is eager to take it to Fury's hometown of Manchester.
Parker announced today he was cutting short his Christmas holiday and returning to his Las Vegas camp to prepare for wherever and whenever the fight may be scheduled.
But he made it clear he preferred for it to be in New Zealand, hinting he would sacrifice some of his purse to make that happen, and hoped the venue decision would not reach the bidding stage.
"I feel it's important that Duco Events and the promoters overseas should come up with a negotiated plan, before it gets to a purse bid," said Parker. "I feel it's important to have it locked into place before I go overseas or sometime soon.
"Both boxers want more money, so it's just about getting the right deal.
"Home advantage would be more important [than money] and giving New Zealanders the experience of coming to another title fight."
Duco co-owner David Higgins said his organisation was currently working on budgets to calculate how much it could submit as a bid to the WBO.
"The Fury side will be doing the same thing and then it all happens in Puerto Rico next week, when the open the envelopes.
"Whoever pays the most, owns all the rights to the fight and whoever loses has no rights, so the stakes are high and, as Joseph says, we do value home advantage."
But Parker also insisted that he would make the most of any scenario, including the prospect of a road trip to northern England.
"Most of my fights have been here in New Zealand, but I've fought in America twice and fought in Germany and I've fought in Samoa, so it doesn't matter where the fight is, I'll be comfortable.
"[Fighting in Manchester] would be a new experience and I'm sure it would feel different to here in New Zealand, with everyone supporting me, but if I train hard, I'm comfortable wherever it is."
It could even be useful preparation for bigger fights to come, as he tries to unite the four major world heavyweight titles. It's almost inevitable those bouts would be staged in England, where the fight game is very popular.
Higgins pointed to the upcoming Anthony Joshua v Wladimir Klitschko encounter, scheduled for April and expected to draw a sellout crowd of 85,000 to Wembley.
"British fans go to everything," he quipped.
"They'll go to the opening of an envelope."
After enjoying the Christmas break with family and friends, Parker joked that he now needed to drop some weight in preparation for a Fury showdown.
"I've put on a bit - I love those burgers and pies - but it's time to get back to work and focusing.
"I'm probably weighing 115-116kg, so would need to lose 5-6kg, but I haven't jumped on the scale yet. It may be more, who knows."
His return to training means he will be unable to attend two key events in his calendar - the Halberg Awards, where he is a finalist as Sportsman of the Year and Favourite Sporting Moment, and the NRL Auckland Nines, where he was due run water for the NZ Warriors.
"Sorry to the Warriors boys," said Parker. "I said I'd be there to support them and encourage them to pick up that trophy, but I'll watching from overseas."
