Fury offered his congratulations to the WBA and IBF champion from his training base in Marbella, Spain, but is adamant he would regain two of his old world titles by inflicting a first professional defeat on "AJ".
Joshua, 27, stopped Klitschko in the 11th round of last Sunday's (April 30) Wembley super-fight to add the vacant WBA and IBO titles to his IBF crown.
Eddie Hearn, Joshua's promoter, then downplayed any suggestion that the bout would take place any time soon.
"It's the heavyweights that make boxing big".
Joshua said his post-fight treat of apple crumble had disappointed him, saying he preferred a "cheap Tesco one".
"It was a great fight", Tyson told ESPN.
"The heavyweight division is always about your longevity because I don't know a heavyweight in any era that didn't have a chance of knocking you out in one punch".
'Hands behind my back, touching the top of my head, making a right mug of him.
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Joshua dismissed the jibes as "competition talk", and explained that Klitschko's knockdown of him actually helped spur him into action during the second half of a scintillating contest. And while the former "Baddest Man on the Planet" is a keen follower of the UFC - noting bantamweight champ Cody Garbrandt as one of his favourite fighters to watch - he believes the balance of power is tipping back in boxing's favour.
"I had a fantastic experience in specifying this Range Rover, and the level of design detail in the finished vehicle is stunning, from the boxing glove emblems in the door handles to the custom storage unit for my training equipment". I love that he goes out there to hurt his opponent, too.
"It's hard to stay [nice], believe me", Tyson said, claiming Joshua will soon ambandon his ambassadorial image without even realising it.
I was talking to him and I told him, "If you let me get through this round, I'm going fuck you up".
"I thought Joshua was getting knocked out in the fifth round, I was nearly crying, I thought he was gonna cost me 20 or 30 million quid", added Fury. That's what boxing is all about but most fighters just don't have that hunger.
"I really don't think Deontay Wilder would be a good fight for Joshua, I'd much prefer to see him fight Mike Perez".
"I could see him fighting in the Bird's Nest Stadium (Beijing) - and Africa".
"Of course we have a rematch in the contract".
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