In a video posted to Twitter, Fury said that he will be fighting soon and in the next few months and said that he needed to call out Josua "just in case".
He added: "Anthony Joshua, I know you have just lost a fight to Usyk and you're beltless at the moment. I'd like to give you an opportunity to fight me for the WBC Heavyweight championship of the world, and the lineal championship, in the next few months.
"You are 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 will send you the date over and we can rumble. A 'Battle of Britain' for the WBC title. Let me know if you're interested, if not, I will select another opponent."
Anthony Joshua responded via Twitter later in the day, saying he would be ready in December.
Talks between Fury and Usyk had been ongoing, with the Ukrainian, according to Fury, not wanting a fight this year.
In an interview that took place during World Wrestling Entertainment's (WWE) 'Clash at the Castle' event in Cardiff last weekend, Fury told reporters that Usyk did not give him an answer about a unification title fight before the deadline of September 1st - a deadline Fury had set himself.
"Usyk was calling me out after his last fight with [Anthony] Joshua, I've replied and said let's do the fight this year, wherever they want to do it.
"I've been waiting for offers from countries to come forward, and all of a sudden Usyk has stated he doesn't want to fight anymore, he wants to fight next year, not this year
"So I'm not going to wait around for anybody, I'm announcing a fight next week."
In an interview via an interpreter with Sky Sports, Usyk said on Friday: "He wants to fight against me. He wants people to talk about him.
"He hasn't retired yet, it's just a game - 'I'm leaving boxing', 'I'm coming back'. He still wants to earn money."
Now, the self-styled 'Gypsy King' is insistent he is not waiting around for Usyk, instead challenging Joshua for a bout that has been hotly anticipated for years.
Joshua and Usyk are not the only fighters that Fury has called out recently. He has also added that he wanted to finish off a trilogy of bouts against another Brit, Derek Chisora (38) following on from his other three-fight tussle against Deontay Wilder (36).
