"(Tyson) Fury where you at, baby?" Joshua said while still in the ring.
Fury has responded to that call by pledging that he would reclaim two of his old titles by inflicting the first professional defeat on WBA and IBF heavyweight champion Joshua.
"Styles do make fights but I am sure I can beat AJ with one arm tied behind my back," Fury said in a Sky Sports interview.
"'I don't even need a warm-up if he wants this.
"I have been out of the ring as long as Klitschko but the difference is, I am not 41, I am 28."
Fury has not fought since winning a unanimous decision over Klitschko in November 2015. The self-styled 'Gipsy King' surrendered the various titles he won in that fight to focus on his mental health problems.
Fury subsequently had his licence temporarily revoked by the British Boxing Board of Control.
He must wait to see if his licence is returned at a hearing next week, but Fury said from his training base in Marbella that he was delighted Joshua beat Klitschko.
"It was an excellent fight, very entertaining and enjoyable and I was screaming for AJ to smash him," Fury said.
"I wonder what people would be saying today if Klitschko had done him in the sixth.
"I was screaming, pulling my hair out - or what little hair I've got left - because I was worried it would cost us millions!
"He showed a lot in that fight. He showed he can get dropped and come back, which is what champions are made of.
"He showed he can recover from taking big shots.
"There's only one fight out there, the biggest fight in the world and everyone knows that. It is the heavyweights, it is me and AJ, no one else.
"It is the only one the world wants to see and I am here, I am the lineal champion, I am still number one in the world and everybody knows that.
"We all saw (his career) had a life and death situation against Klitschko, but Klitschko couldn't land a glove on me."
Fury criticised Joshua's muscular physique, describing the 27-year-old as "a pumped-up weightlifter".
"AJ did really well to come back but I am not 41 or getting old," Fury said.
"He is just a pumped-up weightlifter but (Klitschko) couldn't land one of those of those hooks on me, at all.
"I would just like to congratulate AJ for getting through the nip-and-tuck, life-and-death situation, against an old man, so well done.
"But let's face it, I am the man, the number one in the division.
"There can only ever be one heavyweight, especially in our part of the world, and it's me. It definitely isn't a pumped-up heavyweight, I can tell you that."