Fury has made several U-turns on his decision to retire after a sixth-round stoppage of Dillian Whyte in April, and has been widely touted to be Usyk's next opponent in the wake of his split-decision win over Joshua in Jeddah last month.
But the Ukrainian is reportedly ready to wait until 2023 for his next bout, leading Fury to vow he would "wait around for nobody.
Fury, who has 32 wins and one draw from his 33 professional fights, promised to announce a new opponent within the next few days, and the 34 year-old appears to have his sights set on Joshua.
"Tyson Fury here, aka the Gypsy King, the WBC heavyweight champion of the world, as you can tell," Fury said in a video posted to Instagram, wearing his belts.
"I think you've all heard that I'm going to be fighting soon, within the next few months, and I think that before I announce an opponent, I need to do this, just in case.
"Anthony Joshua, I know you've just lost a fight to Usyk and you're beltless at the moment, and I'd like to give you an opportunity to fight me for the WBC heavyweight championship of the world in the next few months.
"You're coming off a 12-round fight, so you're match-fit, you're ready.
"I'm giving you a few months notice, if you're interested I'll send you the date over and we can have a rumble, a battle of Britain for the WBC championship.
"Let me know if you're interested. If not, I will select another opponent. Thank you very much, and good night."
A bout between Fury and Joshua has been rumoured for several years. Should it be arranged, it would represent a huge chance at redemption for Joshua, who had lost three of his last five fights after winning each of his first 22.