Joshua lost his WBA, IBF and WBO heavyweight titles as Ukraine's Usyk secured a unanimous decision win at Tottenham Hotspur Stadium on Saturday.
Earlier this year it had appeared Joshua was set for an all-British showdown with Fury, the WBC champion, before before a court ruling in May saw the latter ordered to have a third bout against Deontay Wilder.
"Never worry about the 'Gypsy King' fulfilling his end of a bargain – I will always f****** fight until there's not a fight left in me. You just worry about the other people doing their end of the bargain."
Fury vowed to "splash Wilder to bits" and said he was feeling "a million cent", adding: "I'm in fantastic shape, fit as a fiddle.
"I'm absolutely ready, today, tomorrow and forever. I'll always be ready and I'll never make excuses.
"When I beat Wilder, I'll be on to the next one, so on and so forth. It's never about the opponent. It's the Tyson Fury show until I hang those gloves up.
"Until that day, it's all about me, and the roadshow continues. All these years, 2008 to 2021 and I'm still undefeated.
"There ain't a man out there born from his mother that can stop me or beat me – I haven't seen one yet anyway. Maybe he's not born, or maybe he is but he hasn't got the guts to come and fight me."