The 34-year-old looks to have reversed his decision to retire, consistently calling for a pay packet of $500million (£424m) to fight Oleksandr Usyk in an undisputed heavyweight title clash.
Usyk beat Joshua to retain his WBA, IBF, WBO and IBO titles in a split decision victory in Saudi Arabia last month and subsequently called out Fury, who at that point claimed to still be retired.
Following the U-turn, Fury has urged Usyk to fight this year, although the former has pushed back on that timeframe.
While Joshua is coming off the back of two consecutive losses, any meeting with Fury would likely be the biggest fight in British boxing history and the allure is difficult to ignore.
In a video posted on Twitter, Fury said: “I think you’ve all heard that I’m going to be fighting soon within the next few months.
“I think that before I announce an opponent that I need to do this just in case.
"Anthony Joshua, I know you’ve just lost a fight to Usyk and you’re bout-less at the moment, and 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’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 rumble. A battle of Britain for the WBC heavyweight championship of the world.
"Let me know if you’re interested, if not I will select another opponent."