The pair hold all four of the major world titles in boxing's blue riband division and speculation about a contest has been steadily building, reaching fever pitch this week following announcements from the camps of both fighters.
Here, the PA news agency looks at the latest developments and whether they bring a much-anticipated all-British bout closer to fruition.
What has happened this week?
Joshua and Fury's handlers have been in discussions for weeks concerning a prospective blockbuster showdown between the pair. And Joshua's promoter Eddie Hearn was comfortable enough to tell Sky Sports on Wednesday: "Joshua and Fury have agreed to a two-fight deal, in essence." One of Fury's promoters, Frank Warren, said later on: "Things are going in the right direction and we couldn't be more happy that a potential mega fight is in reach."
Any word from Joshua or Fury?
Shortly after Hearn had broken the news, Fury ensured excitement reached peak levels as he confirmed he and his rival could share a ring to determine the undisputed world heavyweight champion in 2021. Fury, who dethroned Deontay Wilder to become WBC heavyweight champion in February, said on Instagram: "The biggest fight in British boxing history has just been agreed. Two-fight deal. Tyson Fury versus Anthony Joshua next year."
So when and where will this happen?
As Hearn pointed out, there are number of obstacles to overcome, with no contracts signed and dates and venues yet to be determined – although all parties seem united in that the first meeting will take place next year. That is predicated on Joshua first overcoming mandatory challenger Kubrat Pulev, while Fury is contractually obliged to face Wilder for a third time. Those fights were scheduled to take place this summer but have been shelved because of the coronavirus pandemic.
Is there anything else that could throw a spanner in the works?
Joshua needs to overcome Pulev first but Oleksandr Usyk lies in wait as the WBO's mandatory challenger. However, an undisputed title fight means that could be put back. Complicating matters on Fury's side is Dillian Whyte, who has indicated he would not be willing to wait for a title shot the WBC has said he must receive before the end of February next year. But with Wilder currently convalescing after bicep surgery, Fury-Whyte could take place before then.
So, where could the first fight be held?
Another potential sticking point. Joshua received criticism for fighting in Saudi Arabia last December when he outpointed Andy Ruiz in their rematch to regain his status as world heavyweight champion but the country is the bookmakers' favourite to stage his opener against Fury. Las Vegas – where Fury defeated Wilder – should never be discounted as an option while Wembley is lower down the list. The public health crisis and whether fans are allowed back into stadiums next year could have some bearing on where the fight takes place.
But this is still positive news, right?
As Hearn said: "The most difficult part of any deal is the financial element." It seems Joshua, the WBA, IBF and WBO champion, and Fury, who holds the WBC title and the prestigious Ring Magazine belt, are on the same page in that respect. It is understood there would be an even 50-50 split the first time around while the winner would pocket 60 per cent of revenue in any return fight, extravaganzas which Hearn predicts will never be topped in this generation, adding: "It's the biggest fight ever in British boxing."