Talks between Oleksandr Usyk, the WBO, WBA (Super) and IBF champion, and WBC title-holder Fury broke down last week.
Joshua, 33, faces American Jermaine Franklin on Saturday in his first bout in 2023.
He says he wants an all-British fight with Fury, whatever the outcome against Franklin at the O2 Arena.
Joshua said: “There’s no better time to get Fury in the ring than now because he needs me to redeem himself from this circus, this letdown.
“He needs me so there’s no better time than for him to call my name out and I’m someone that will take on any challenge.”
Joshua was in talks with his British rival over a fight last year following successive losses to Usyk, but could not reach an agreement and Fury fought Derek Chisora at Wembley Stadium in December instead.
Fury then set his sights on facing 36-year-old Usyk in what would have been the first undisputed heavyweight contest since 1999.
But after weeks of negotiations, both parties announced an agreement could not be reached due to a dispute over the terms of a potential rematch.
“It’s not my position to slate or slag anyone off. I’m pretty sure that fight [Fury v Usyk] will happen because I feel like there could potentially be a method to the madness,” Joshua told Steve Bunce on 5 Live Boxing.
“If, number one, I was in that position [and] didn’t take that fight, how would people treat me?
“And number two, if I didn’t take that fight what would Fury have to say about me? So I’m not going to stoop to that level.
“I respect everyone that steps into the ring. But I just sometimes think some people should watch their mouths a bit.”