With Fury set to return to action for the first time since ending Wladimir Klitschko's heavyweight dominance in November 2015, many are billing the prospective meeting with current WBA Super and IBF world champion Joshua as a simple match-up of boxer vs. puncher.
According to reports, the contest will take place on 31 March, at Cardiff's Principality Stadium.
Contracts are expected be signed and a statement will then be released by the fighters' respective promoters, Matchroom Boxing and Duco Events.
Parker's representative, David Higgins, has been in the United Kingdom since Monday to conclude negotiations, and a news conference is expected next week.
Although the fight has not been made official, Higgins, who arrived in London this week to finalise a deal with Joshua's promoter Eddie Hearn, said an announcement is imminent.
"We are prepared to sit down and make that fight now with Tyson Fury and Anthony Joshua is as well, because he called me straight after Tyson was texting him the other day". "It's not even in question for them", he told Sky Sports.
Anthony Joshua will choke against Joseph Parker, says David Higgins
"Let me tell you, that's a fight that we would sit down now and make for 2018".
"We have a hypothesis that Joshua is like the English football team".
And Hearn recently outlined Joshua's plan for the year, which involved beating Parker before taking on Wilder in an attempt to hold all four heavyweight belts. David Higgins has got a lot to say for himself, he's a amusing character, and Anthony Joshua doesn't like what Joseph Parker has been saying.
"He asked me to announce a fight in April and that's what we're working on, a defence of his Ring belt".
Fury, 29, has been out of action for over two years and still has licensing issues to deal with as the former world champion bids to prove his fitness and mental state to the British Boxing Board of Control. But Parker has won all 24 of his professional fights.