It comes after it was revealed this week that Fury's team have been in talks for a unification bout with Usyk, holder of the WBA, WBO and IBF belts with the Ukrainian's team, with negotiations ongoing since November last year.
Moreover, the respective teams have not come up with the requisite step-aside deal for Joshua - believed to be over £15 million and a guarantee to face the winner - to agree to sign off on the process.
Seemingly turning his focus to Whyte, Fury said on Friday: "I can't wait to punch Dillian Whyte's face right in. I'm going to give him the best hiding he's ever had in his life, boy. Dillian Whyte, train hard sucker 'cos you're getting annihilated, bum."
Fury also accused Joshua and promoter Eddie Hearn of being "the worst businessmen in the world" for not concluding the tabled step-aside deal, with Joshua entitled to a contractual rematch against Usyk after losing three of the world title belts to the Ukrainian on September 25 at Tottenham Hotspur Stadium.
The self-styled Gypsy King claimed Joshua and Hearn had "lost $90million in a day" in reference to Joshua's step-aside fee and guaranteed purse to meet the winner, though Hearn insisted Fury's demands had stalled an agreement.
"You never know what he [Fury] wants from one day to the next but at the moment he is insisting on a warm up fight with his mandatory challenger Whyte which would delay everything.
"AJ can't wait around all year while Fury goes on to meet Usyk in late summer. He would need contracted cast-iron guarantees that he would get his chance of becoming the undisputed champion by December at the latest," said Hearn.
The complexity of the talks also extend to Whyte, who has the right to challenge for the belt held by Fury. The purse bid situation has also been further nuanced with Whyte having disputed and subsequently appealed the 80/20 purse split in favour of the champion ordered by the WBC.
And furthermore, Whyte is also locked in legal battles with the WBC over his status and lack of opportunity to fight for the belt going back two years. That, too, is likely to need to be resolved before Fury and Whyte meet in the ring.