The IBF and WBA 'super' heavyweight champion is first plotting a November return, and is awaiting to hear if Wladimir Klitschko accepts a rematch.
I listen to what people want. Tyson makes a lot of noise and he's made it clear that he wants to fight me.
But Fury is still Joshua's top target, and the 27-year-old told Sky Sports News HQ: "I listen to what people want. Tyson makes a lot of noise and he's made it clear that he wants to fight me.
"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. I've built a relationship with the United States."
Asked if it was logistically possible to hold all four major heavyweight titles, Joshua said: "You can never set your standards too low, let's aim high. I will state today that it's very possible."
After climbing off the canvas to stop Klitschko at Wembley Stadium in April, Joshua is hoping to welcome his veteran rival back to the ring later this year in a rematch sanctioned by the IBF.
"I would say late-October or the middle of November because early-December is the last time [this year] that we can fight.
"So let's say the middle of November, sometime. That gives me three months to train. Early-August, I'll start my training camp to fight Klitschko or Kubrat Pulev."
Bulgaria's Pulev remains the mandatory challenger to Joshua's IBF title, so will be the Brit's next opponent if a rematch with Klitschko doesn't come to fruition.
"The exemption is that if I do box Klitschko, I can keep [the IBF belt]," Joshua explained. "But if Klitschko doesn't take the rematch, I have a challenge lined up by Pulev, who is a great fighter who has challenged for a world title before.
"I'm now at a level with guys who are either champions or have fought for a championship belt. Pulev lost against Klitschko after the training camp when I sparred with Klitschko."
Promoter Eddie Hearn has already discussed the possibility of staging a rematch against Klitschko in Cardiff's Principality Stadium, Nigeria or Dubai.
"Cardiff is a good option - same time zone, I train locally, and I know how to prepare myself on home soil," Joshua said. "Africa sounds good because it brings prestige, like when Muhammad Ali fought George Foreman in the 'Rumble in the Jungle'.
"Everybody likes Dubai because you get the sun. But what is feasible for the majority of people, not just a small percentage?"