On Tuesday, Fury has appeared to suggest he will have to fight Deontay Wilder for a third time after claiming the American had demanded $20million to step aside and pave the way for a showdown with Joshua.
And on Wednesday, Joshua attacked Fury on Twitter, branding his heavyweight rival a “fraud”.
Joshua said: "@Tyson_Fury the world now sees you for the fraud you are.
"You’ve let boxing down!
"You lied to the fans and led them on. Used my name for clout not a fight.
"Bring me any championship fighter who can handle their business correctly."
Joshua tagged Fury in the post, and the latter fired back immediately, “@anthonyjoshua your [sic] more full of s*** that Eddie, Spouting absolute s****! your full team knew there was an Arbitration going on, it was out of my hands! but i tell you what if i’m a fraud let’s fight this weekend bare knuckles till 1 man quits? let’s put up 20 mill each!!!”
Joshua hit back as the pair traded tweets, questioning why Fury would announce they were fighting if the arbitration was in process.
“If there was an arbitration going on, why announce to the world we are fighting! The fight was signed! UNDISPUTED,” Joshua added.
“Bare knuckle? You’re a good kid, don’t play with me Luke! I’ll slap your bald head & you’ll do nothing! Waste man.”
Fury insists he will take out Wilder as well as Joshua.
Fury added: “Not going in to the details on line!
“You’ll slap me about will you dosser please come and try Bum, i’m waiting... Femi AKA Bottlejob! 24/7 365. Ready. i’ll smoke wilder first then you will get yours aswell.”
The fight between Fury and Joshua was thrown into doubt by a legal ruling in the United States that Fury is contractually bound to fulfil a third meeting with Wilder.
The decision by an arbitrator, which can be legally binding, came just hours after Fury appeared to confirm his clash with Joshua would take place on August 14.
While the decision would not necessarily scupper a Fury-Joshua deal, it would require Fury's team to summon a substantial pay-off in order to convince Wilder to temporarily step aside.
And it would also seemingly scupper hopes that the deal struck by Fury and Joshua could include a clause for an immediate rematch between the pair.
Fury and Wilder fought a split-decision draw in December 2018 in Los Angeles, before Fury won their second meeting by seventh-round stoppage in February last year.
Attempts to secure a third fight were complicated by an injury to Wilder and an absence of available television dates, prompting Fury to move on based on the assumption that any agreement had expired.
Following the announcement of the arbitrator's decision, Fury posted on his Instagram story with a message across a background photo of Wilder with new trainer Malik Scott.
"What a joke bronzebomber has become. Asked for 20 million to move over £joker," Fury wrote in the post.
"Looks like I have to crack his skull again."
Fury had announced earlier this week that the unification clash was "100 per cent on" for August 14, adding: "I cannot wait to smash Anthony Joshua on the biggest stage of all time."
Speaking on Matchroom's social media channels on Tuesday, Eddie Hearn had stressed the need for Fury's camp to find a swift resolution.
"We can't wait around," Hearn said. "We had a deal in place with Tyson Fury and we were told the arbitration wouldn't be an issue, that we could move on with this fight.
"They were wrong and that's on them, that's their responsibility and their problem.
"We hope they can solve that problem, but we have to look after ourselves and Anthony Joshua.
"We have to maintain the position of unified world champion, and those talks will continue and we want to be in a position by the end of this week to know, are we fighting Tyson Fury or are we moving forward with another option?"
Additional reporting by PA.