The British heavyweight, holder of three of the four world belts, has been accused by some fight fans of ducking the big match-ups, while rivals Wilder and Fury had a thrilling brawl in Los Angeles at the weekend.
Fury called Joshua a "chicken" and a rematch between him and Wilder looks inevitable in the wake of their controversial draw.
That could mean Joshua being frozen out of the division's top fights for a little while yet, but the 29-year-old reckons he is ready to go.
After Wilder's co-manager Shelly Finkel suggested a Wilder-Joshua showdown could happen, Joshua wrote on Twitter: "What took this fool so long? Like we ain't been interested?!
"Anyway well done Fury! They (Wilder) wanted to get to you because they assumed you was finished!! I'll give you a fair one when your ready. Either one of you!"
Joshua's promoter, Eddie Hearn, has gone to Los Angeles this week and says he is hoping to talk about an April date between his fighter and Wilder.
Fury's promoter, Frank Warren, however, hopes to open talks over Fury-Wilder II.
Hearn told the Daily Mail: "First we need to find out what is happening with a Wilder-Fury rematch and whether the rematch clause we are hearing about is what it seems.
"If anything, the way that fight went on Saturday works in our favour in terms of getting Wilder. That has always been the fight we want because he has the last belt.
"It might also be the case that Wilder fancies his chances in a punch-out with AJ rather than another go at Fury. As far as AJ is concerned, he will fight anyone. He would love Wilder next and he has always wanted Fury.
"I am going out to LA and we will talk (with Wilder's representatives) this week. It is a fight we definitely want for April and will work hard to make."
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