Fury, 28, has not fought since dethroning Wladimir Klitschko as world heavyweight champion in November 2015 because of drug issues and mental health problems.
But he is back in training and following Joshua’s epic victory over Klitschko at Wembley last month, Fury says Joshua’s promoter Eddie Hearn is working to organise a bout between the pair.
“Eddie Hearn has already booked Wembley Stadium for next April,” Fury told Monday’s edition of The Daily Telegraph.
“There isn’t another stadium where it should take place. I would fight Joshua in October, but I believe Klitschko will take the rematch.”
Joshua, 27, stopped Klitschko in the 11th round of their bout to add the vacant WBA and IBO titles to his IBF crown.
He has offered Klitschko a rematch, but says he also wants to fight Fury.
Fury hopes to make his comeback in a fight on July 8, but must overturn a suspended drugs ban at a UK Anti-Doping hearing that was due to begin on Monday.
The Manchester fighter has admitted to taking cocaine to help with depression, but denies taking any performance-enhancing substances.
“I’ve never taken a (performance-enhancing) drug in my life,” he said.
“These people can say what they like, but I’m suing them for what they have done to me. They thought they had a villain and that they were going to take me down.
“This went on from before the Klitschko fight. If I tested positive then, why didn’t they ban me then?
“To be honest, I didn’t want to live. If I had cheated, fine. But I have not cheated. I beat Klitschko on ability. I don’t need a drug. I’m a giant already.
“For them to say that about me, it sent me under. I did not want to live anymore. But I feel differently now.”