"I'll give you a fair one when your [sic] ready!"
The brit holds the WBA, WBO, IBO and IBF titles.
Fury's promoter, Frank Warren, however, hopes to open talks over Fury-Wilder II.
Joshua has a Wembley date booked in for April 13, 2019, but with no opponent named as yet.
He 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".
The Olympic champion has responded to Wilder's comments that he is "very interested" in a colossal clash with AJ, and said he is ready whenever to go at it with the American and blames him for the lack of a fight in the past.
However, the fight never materialised, and Wilder took on Fury in Los Angeles instead.
"If everything is in the paper, I have no desire to discuss anything, and let him go fight who he wants".More news: CCTV shows crime chaos at Sasol garage after Global Citizen Festival
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If Wilder's explosive power already posed a significant threat, the biggest question surrounding Fury's chances was whether he had truly recovered from that period, in which he admitted to taking cocaine, becoming suicidal and lived to such excess he reached an estimated 27 stone. As far as AJ is concerned, he will fight anyone.
Listen 100 per cent we'll do the rematch, but we're going to go away, recalculate, see what's going to happen, get back together, said Fury. "It is a fight we definitely want for April and will work hard to make". I don't know what fight the judges were watching.
But Joshua has hit back at Finkel's claims and has suggested that Wilder wanted to face Fury because it would be an easier fight. "Now, Fury's the people's champion".
There was also an offer to fight Fury in Joshua's tweet.
He was expected to face Dillian Whyte in a rematch of their 2015 classic, but the silver WBC titlist will now face Dereck Chisora on December 22.
"This event brought more pressure than I expected".