The blockbuster bout is likely to take place between next May or June after a breakthrough in negotiations.
The Matchroom promoter Hearn admitted that ‘minor issues’ need to be resolved before an official date is confirmed.
It is set to be one of the biggest British showdowns ever as the two giants to head-to-head to determine who will be crowned undisputed heavyweight champion of the world.
The Atheltic’s Mike Coppinger wrote on social media: “Eddie Hearn says Tyson Fury-Anthony Joshua is verbally agreed to and contracts are being drafted for the heavyweight championship fight in May/June.
“Hearn says only ‘minor issues to resolve’ such as who walks last, the site, etc. This fight is going to happen next.”
It is likely to be a two-fight deal worth a staggering £500million while the destination could be in the Middle East.
AJ overcame his first obstacle by breezing past Kubrat Pulev last weekend while Fury remains in a legal battle with Deontay Wilder over their failed trilogy of fights.
Another obstacle preventing the deal is the fact Oleksandr Usyk is currently the mandatory challenger for AJ’s WBO belt.
Hearn has previously admitted that both fighters could ditch their belts, from the four major sanctioning bodies, if they try to block the undisputed decider.
He said: “We’ve just got to be a bit careful because you may be on the verge of saying: ‘Let’s just get rid of politics right now.
“Let’s just drop all the belts because if we lose one belt, we might as well lose them all. We will do whatever we can to make sure that every belt in boxing is on the line in the Tyson Fury fight.
“What we won’t be is held to ransom and we won’t let politics get involved in a fight that boxing needs.”