Fury, 34, had said he was retiring after beating Dillian Whyte in April, and reiterated that claim last month, before challenging former world champion Joshua in a video posted on social media.
Joshua was beaten by Ukrainian Oleksandr Usyk in their rematch in Saudi Arabia last month, his third defeat in his past five bouts.
There is a rematch clause with a 50-50 split if Joshua beats Fury, but the venue for the December fight, should it go ahead, is yet to be confirmed.
Cardiff’s Principality Stadium has been mooted, while Saudi Arabia or another country in the Middle East could also host it.
There are still hurdles to overcome before the fight is finalised, with both fighters signed to rival promoters and broadcasters, although two-time champion Joshua this week entered the WBC rankings in sixth position in another significant development.
A boxer must be ranked in the top 15 by all governing bodies to challenge for a world title. (BBC Sport)