Fury claimed he was retiring after beating Dillian Whyte (34) in April but appears set to perform the latest of several U-turns over his future in boxing.
The 34-year-old WBC belt holder is now searching for a high-profile opponent after rival champion Oleksandr Usyk (35) said he does not intend to fight again until 2023.
In a bid to tempt Joshua to get in the ring with him, Fury posted a social media video on Tuesday in which he said he had left the 32-year-old with "no excuses" not to accept the bout.
"I'm being bombarded with messages on how much I'm going to pay AJ - everyone is saying 80-20, 70-30, 75-25," Fury said.
"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 or 30 per cent. I've offered (his) people 40 per cent - take it or leave it."
Joshua suffered a second straight defeat to Usyk in Saudi Arabia last month.
Joshua's promoter Eddie Hearn had expressed scepticism about Fury's initial offer on Monday, but said he was willing to discuss it with his counterpart Frank Warren.
Hearn previously said he was considering options for a return for Joshua early next year, a decision which could change if Fury is true to his word.
"Fury's outstanding at getting the public to believe absolute nonsense," Hearn said on talkSPORT.
"It worries me actually what people do believe, if you look at the timeline over the last two or three weeks, it's unbelievably bizarre.
"Personally, I don't believe Fury is serious about this, but if he is, which I said to the Warrens last night, 100 per cent we will sit down and make the fight.
"I would love to get really excited about this, because it's the fight that I get stopped in the street for more than any fight that could be made and it's a fight that AJ has wanted for a long time."
smg/mw
