Joshua added the WBA and IBO belts, vacated by Fury who is without a licence amid a hearing into alleged anti-doping offences, to his IBF strap with a thrilling 11th-round TKO of Wladimir Klitschko.
He said on Tuesday that Fury is top of his list and his British compatriot, who claims he is still the lineal champion, duly responded to those comments, saying in an Instagram post that "the world wants to see it" but that he does not believe Joshua can lace his boots up.
But responding on Sky Sports News, Joshua said: "I don't think Tyson can lace his own boots at the minute! When Tyson is fit, when he's ready, we'll be ready.
"We're patient. I wish him a speedy recovery with all the stuff that he's got going on outside the ring.
"It's a fight that people want to see. He's one of my boxing rivals and I need this competition throughout my career. I can't wait for him to get back. We'll see if I can lace his boots when we come to blows.
"Charles Martin called me out, I had to beat him then defend my title. Then Wladimir Klitschko wanted to fight me for the WBA, and I put up my IBF, so every fight I've been in was a challenge. It was never given to me.
"I'll always fight Tyson Fury, not because of the 'lineal champion' business. But because the public want to see it. I fight for entertainment, not for belts or glory. Just pure entertainment and passion."
Joshua has yet to receive word back from Klitschko about a rematch and added: "I'm being patient. I've got myself into training to prepare, if it does happen. He's probably looking at the landscape - what is there to achieve?
"What does he really want to do? Does he want to go through that type of fight again? I definitely do.
"He was at the peak of where he could be at the time, so it's interesting to see how much he could change. I went in as an underdog, in the sense of experience.
"I gained so much experience from those 11 rounds so hopefully it will shape me into a better fighter and I can get a better result, if we do the rematch."
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