The two Brits are set to finally clash in 2021 to attempt to unify the heavyweight division.
There have been lengthy negotiations between the two parties for some time but the WBA, WBO and IBF champion has suggested things could be over the line soon.
"Another positive meeting with @258mgt," he tweeted. “They've informed me things are shaping up nicely. They will be sharing the update with Eddie and you'll be hearing from me soon."
Joshua last fought in December, defeating Kubrat Pulev at Wembley, while Fury hasn't been in the ring since knocking out Deontay Wilder more than a year ago.
A number of elements still need to be ironed out before contracts can be signed with the venut for the fight itself still to be decided.
Eddie Hearn recently revealed how the global coronavirus pandemic has made securing an appropriate country to host the clash even more complicated.
“A lot of the dates now do depend on the Covid situation because originally we wanted it to happen in May but now we feel like June is a safer option. Different countries are at different stages. Some countries are in lockdown, some are open.
“Saudi Arabia are in lockdown but they are staging major events behind closed doors. I believe they plan to open up in March but you just don't know what's going to happen.
Another positive meeting with @258mgt. They've informed me things are shaping up nicely. They will be sharing the update with Eddie and you'll be hearing from me soon 👊🏾
— Anthony Joshua (@anthonyjoshua) February 23, 2021
“Las Vegas will be opening back up in May and it would be a huge opportunity for them to get the biggest fight in boxing. Singapore, Qatar, Abu-Dhabi, Dubai. All these countries want to stage this fight.
“I wouldn't mind a trip to Vegas in June, I think it would be a great home for the fight, but we can't ignore the seismic offers from other areas of the world.
“All I can say is we're finalising the minor details and nothing major that we see is going to stop this fight from happening.
“There's no danger that this isn't the next fight. I don't know what we'd do if we don't fight Tyson Fury. Everyone is so focused to get this over the line.”