Joshua won back his WBA, IBF, WBO and IBO belts from Mexican-American Andy Ruiz Jr in Saudi Arabia last December.
Joshua is yet to have his say on the upcoming fights, leaving promoter Hearn - who claimed talks were progressing smoothly - to reveal the news.
Joshua's promoter Eddie Hearn has been negotiating with Fury's management for several weeks and he said the terms were in place for the fighters to go head to head. It's fair to say (Joshua and Fury) are in agreement regarding the financial terms of the fight. And it is from Fury's side, as well.
"We're in a good place", Hearn said.
"For one, we don't know how this will effect pay-per-view buys and it's going to be hard to replace what we had in gate receipts". 'It doesn't get bigger, and there will never be a bigger fight in our generation.
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Heavyweight boxing could have a unified champion for the first time in more than two decades.
The Fury-Wilder rematch drew a sellout crowd in Las Vegas that set a record of more than $17 million for the live gate, which is separate from pay-per-view. We have agreed with Tyson Fury and the team and we can now move forward with finalising the smaller parts of the deal.
The dream bout is sure to happen, but before that, Tyson Fury will need to get past Deontay Wilder in their third fight and Anthony Joshua will need to get past his next opponent, which is mandatory challenger Kubrat Pulev.
In April, Joshua challenged Fury to a fight to determine one dominant figure in the heavyweight division. "Big shout out Dan, he got this done, literally over the line, a two-fight deal, Tyson Fury versus Anthony Joshua next year".
The first of the two fights is likely to take place in the Middle East in the summer of 2021. I believe we're in a great place where both guys have agreed to what that should be.
"The biggest fight in British boxing history has just been agreed", Fury said in a video message posted on Twitter. But obviously there's going to be significant offers, and there already has been to stage this fight, the biggest fight in boxing, one of the biggest fights of all time, and it's for the undisputed heavyweight championship. With Joshua now holding three world titles and Fury in possession of the remaining two in the WBC and Ring Magazine belts, the contest between them is going to be for all of the marbles.