Fury and Joshua have been locked in talks over a long-awaited meeting, but the 'Gypsy King' is impatient with the lack of confirmation, calling his rival a "coward" last week.
The two former heavyweight world champions are expected to meet in November, but the location of the fight has yet to be agreed.
In an expletive-laden video posted on Instagram, Fury, 38, claimed Joshua was pulling out of the fight.
"This will only lead me to a fight with Oleksandr Usyk," he said. "Oleksandr, you're his (Joshua's) trainer and his sensei, and you know he is... weak-minded... and that he would never fight the Gypsy King, which you have had the guts to do twice.
"You got the decisions last time; fair play to you. Good luck to you; let's do it again. Let's run it back in November.
"The ball is now in your court... do you want to fight me this year, 2026, or are you calling it quits? Let me know because I am now looking for a new opponent."
Fury has twice lost to Usyk, a two-time undisputed heavyweight champion.
Joshua has been working with the Ukrainian's training team.
New York's Madison Square Garden was widely expected to be announced as the venue for the Fury-Joshua fight even though Joshua's camp prefers Wembley Stadium.
Sela, the Saudi state-owned promotional company financing the fight, and broadcaster Netflix are believed to favour staging the bout in the United States.
