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Joshua lost to Oleksandr Usyk in a rematch last month, failing to regain his WBA, WBO, IBF and IBO titles from the Ukrainian.
Usyk seemed set to take on Fury for the WBC belt, yet instead it seems a clash between the Brits is more likely.
Fury wants to fight Joshua before 2022 is out, with the latter tweeting on Monday that he would be "ready in December".
On Tuesday, Fury revealed the details of the offer made to Joshua.
"I'm being bombared with messages on, 'How much am I gonna pay AJ?'", Fury said in a video shared on social media.
"Everyone's saying 80-20, 70-30, 75-25. The actual answer is: I've offered him 60-40; 40 per cent of this amazing fight, because I want this fight to happen.
"He doesn’t have any excuses now not to take it. He can't say I've low-balled him and offered him 20 per cent, 30 per cent.
"I've offered the people 40 per cent, take it or leave it. Let us know. Boom."
Fury subsequently tweeted: "He will never get a better offer. Straight WBC Boxing shot, the ball is in your court @anthonyjoshua."