The 27 year-old defeated Klitschko in April via eleventh round stoppage following a back and forth battle and already has Fury on his mind for 2018.
"I listen to what people want. Tyson makes a lot of noise and he's made it clear that he wants to fight me," Joshua told Sky Sports.
"Now, I'm in a position that I can say: 'Tyson when you're back, I'm interested, so let's live up to what people want to see us do'. That's why he's top of my list.
"In the UK, Tyson Fury. Internationally, my pick would be (Deontay) Wilder (WBC champion). I've built a relationship with the United States."
This prompted and afternoon repost from Fury, who stated: "Addressing the 'AJ' call out, you need to fight me, because at the moment I'm still lineal champ," said the 28 year-old.
"(I've) been out of boxing nearly two years so that should make it fair on the others. I really don't believe that 'AJ' can lace my boots up. I want to come back and see how good I am - even with the time out.\
Twitter poat by: gypsyking101
Addressing the AJ call out, U need to fight me cuz ATM I'm still lineal champ! Been out of boxing nearly 2 years so that should make it fair on the others. I really don't believe that AJ can lace my boots up, I want to come back & see how good I am even with the time out. Me vs AJ is the biggest fight in boxing without a doubt! If this fight isn't made it will be like Lennox Lewis & riduc bowe. Now there is new challenges I can motivate my self again. Wilder & Aj & Parker even povetkin theses are the fights I want. Let's make this happen. Let's make boxing great again.
"Me vs AJ is the biggest fight in boxing without a doubt. If this fight isn't made it will be like Lennox Lewis v Riddick Bowe.
"Now there are new challenges I can motivate my self again. Wilder, AJ and Parker (current title holders, and) even Povetkin - theses are the fights I want.
"Let's make this happen. Let's make boxing great again," added Fury.