Such a battle is within touching distance following Fury's stunning seventh-round stoppage of Deontay Wilder in Las Vegas in the early hours of Sunday morning to capture the WBC heavyweight belt.
Here, the PA news agency looks at the permutations surrounding a possible match-up that Joshua's promoter Eddie Hearn has already said he would like to see this summer.
What needs to happen for it to take place?
First and foremost, we have to wait and see whether Wilder will activate the rematch clause in his contract for the second meeting with Fury, with late-March the deadline.
Joshua also faces a mandatory clash with Kubrat Pulev, that at present is understood to be slated for June 20 at the Tottenham Hotspur Stadium. Pulev and Fury, however, possess the same promoter in Bob Arum. As Hearn has stated, it would be "an easy manoeuvre" in getting the Bulgarian to step aside.
As for the WBC's number one contender, Dillian Whyte, he has long waited for his title chance. Although guaranteed his opportunity by the sanctioning body in February 2021, it seems certain he will have to wait for his opening then.
What about the money?
Negotiations over purse percentages could be extremely complicated, as we have previously seen when the clamour was for Joshua to fight Wilder, only for that to fall by the wayside.
A smaller obstacle is television rights, with Fury signed up with ESPN, and BT Sport in the UK, while Joshua is with Sky Sports. Hearn says that while there are "hurdles to overcome" with regard to broadcasting, they are "nothing too much".
Where could it happen?
The most obvious choice would appear to be Wembley, where Joshua has twice fought in the past. There would be a guaranteed gate of 90,000, and would almost certainly be Joshua's preference.
Fury, though, has enjoyed being Stateside, where he has marketed himself to tremendous effect since his comeback, winning over the American fans with his charismatic personality. He has suggested if there is to be a third fight with Wilder he would like to see that at the 72,000-capacity Allegiant Stadium in Las Vegas that is soon due for completion. It also represents a viable venue for any showdown with Joshua.
But, as we all know, money talks in boxing, as witnessed when Saudi Arabia staged Joshua's rematch with Andy Ruiz Jr in early December at the 15,000-seater Diriyah Arena, paying a vast sum to be the host. It has been reported the powerbrokers behind it were also ringside for Fury-Wilder II. Money would be no object in putting on the biggest fight for decades.
How do they compare?
Fury would go into the fight with height, reach and weight advantage over Joshua. The more aggressive fighting style he unveiled against Wilder in the rematch, and under the direction of new trainer Javan 'SugarHill' Steward, now makes him a far more dangerous proposition compared to his previous catch-me-if-you-can technique.
Like Wilder, Joshua knows he possesses the power to knock out any opponent, but being the much lighter man, could he cope with Fury's new-found combative approach? Wilder certainly couldn't. He was bullied from start to finish by Fury, and left drained by his opponent's come-forward, smothering plan of attack.
While AJ is almost identical in stature to Wilder, he is no one-trick pony like the American. He knows how to box clever, as he has shown in the past in his two points victories to date over Joseph Parker and his second battle with Ruiz Jr.
So who would win?
Three respected former professionals are all backing Fury.
Ricky Hatton told the PA news agency: "He is the better all-rounder. He's very versatile. From a distance he can outbox you, is quicker, and the way he wants to mix it up, as he showed against Wilder, he's not found wanting in that department. No disrespect to AJ, but I think he (Fury) beats him comfortably."
Amir Khan has also suggested a Fury win, believing he has a style that "can break anyone down". He added: "He has the speed, the movement, the accuracy. If I was to pick someone, I'd lean a little more towards Fury. He is the one who put Wilder down. I never used to think he had that much power!"
Johnny Nelson is also backing Fury. He said: "Gun to my head, I'd have to say Tyson, and I'm a massive fan of AJ. I'm not saying it's a done deal as AJ has everything to beat him. But looking at the performance, the technique used by Tyson, you'd have to side with him."