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The 'Battle of Britain' clash seemed to be at risk of a collapse after Fury issued a Monday deadline for an agreement to be signed but AJ's promoter Eddie Hearn ruled out such swift action.
That raised concerns that the fiercely anticipated bout could fall through, with Fury threatening to walk away and fight someone else.
Joshua has now moved to calm such fears in a video posted on social media, expressing his intent to sign the agreement.
"I've been signing contracts for years. It ain't in my hands, it is with the legal team, that's why you hire lawyers. You know the history of boxing, make sure you get your legal terms right," he said.
"That's why you have good management and good lawyers. Of course I'm going to sign the contract, it's just with some lawyers at the minute."
A fight between Fury and Joshua would likely set the heavyweight scene for 2023, with the winner then poised to face off against Oleksandr Usyk in a unification bout.
Joe Joyce's victory against Joseph Parker in Manchester on Sunday has set the Brit as the mandatory challenger for Usyk's WBO title, though he may have to wait for his shot.