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".
The 28-year-old has not fought since capturing the WBA, WBO and IBF heavyweight titles with a shock win on points over Wladimir Klitschko in November 2015.
He's an important asset for Under Armour because he's demonstrating crossover appeal - the ability to attract new fans to the sport.
Joshua's 11th-round knockout of the Ukrainian has ensured he is one of the most recognisable sporting names on the planet, but he still has some way to go to reach the stature of the late Muhammad Ali. But it was still considered an important test.
"They want to see Joshua's power and the damage he causes".
'I'm getting down from 26-and-a-half stone, back down to 18-and-a-half, that's got to be impressive on its own.
"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".
Bernard Hopkins says boxing is in "fantastic shape" following Anthony Joshua's thrilling victory over Wladimir Klitschko and believes "the sky is the limit" for the undefeated Briton.
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Fury is not that impressed with Joshua's victory against Klitschko.
"With what Joshua can bring now everyone will want to fight him because he is bringing the money to the table".
The 28-year-old hasn't boxed since beating Wladimir Klitschko in November 2015 and is now without a boxing licence, but he has been training in Spain in an attempt to get back into the ring.
'Let's face it, I've been through depression, life and death situations, to come back against all the odds, turn it all around, get myself back in training.
"I was just exhausted, I fell to my feet, I got up", he said.
"When i dropped Klitschko that roar (from the crowd) lifted my arms up as well".
"When you look at Joshua, I was excited that he won, even though I picked Klitschko because of age". Tyson, though, believes there is another feature of Joshua which could see him become one of the most successful athletes of all time.
"If I have to go through everyone in the division, then great". I like it when I'm supposed to lose, and that's when you get the best Tyson Fury.