Venues and dates have yet to be agreed to, and both English boxers have defenses of their titles to get through for the two fights to happen. Boxing promoter Eddie Hearn confirmed that they both have agreed to a two fight deal.
Heavyweight World Champions Tyson Fury and Anthony Joshua will fight each other in 2021.
"We have the Dillian Whyte mandatory [title challenge for Joshua] which is due before this fight", Hearn added. 'There is still a lot to overcome.
"The point of Fury, Joshua and the teams agreeing to the structure of the deal?"
If Joshua and Fury can hold onto their titles, it would set the stage for an enormous unification bout. I believe we're in a great place where both guys have agreed to what that should be. The fights would be among the biggest that could be made in the sport and clearly the most important bout in the history of the boxing-crazed United Kingdom.
Parker says he's stayed in good shape during the Covid-19 crisis and hopes a fight on home soil isn't far away.
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"Things are going in the right direction and we couldn't be more happy that a potential mega fight is in reach".
The problem, of course, is that they can agree all manner of things but the world of heavyweight boxing - which a year ago saw Andy Ruiz Jnr emerge from obscurity to knock out "AJ" before order was restored in the rematch - is a minefield of promoters, politics, sanctioning bodies and broadcasters. The structure of the deal has been put forward, and agreed to by both parties. "But they're two British guys, so in a dream scenario - especially for British fight fans - that fight would take place in Britain". In short, 2021's could be a very prosperous year for both parties.
But the all-British unification fight will NOT be held on home soil, with Fury not now in possession of a British boxing licence.
"The situation is that we have an agreement in place for how the deal structure for the fight should work over a two-fight deal", Hearn said. He got this done, literally over the line", he continued, "Two fight deal.
Fury won the WBC title in February in his rematch with Deontay Wilder after their first bout ended in a draw.
"(But) both guys are in agreement. They have both fought in Saudi Arabia - Tyson did WWE there, and AJ had his last fight there [December, 2019].