In the post on Twitter, Joshua also offered to help his heavyweight rival to shed three stone ahead of his WBA Super and WBO rematch with Wladimir Klitschko in July.
"I think it's just natural genetics," Fury told Sky Sports News. "I am never going to be Mr Universe and I don't need to be. I am a fighting man.
"You don't see a good-looking pitbull, do you? With its tail docked off, it's ears docked off, and their face in bits like it's been chopped up by a badger.
"That's what I am. I am a natural fighter. I am the Gypsy King. I don't need muscles - and you can't put muscles on your chin. I've beaten plenty of six packs, don't worry about that.
"Whatever they put in front of me, I've got the agility and the style to beat and overcome them. You know why? Because I'm a gypsy and gypsies adapt to any situation and any environment."
Fury and Joshua have been tipped to meet in a heavyweight unification bout later this year.