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The 'Maverick' takes on Floyd Mayweather in an eight-round exhibition in Miami, Florida this weekend, with the pay-per-view bout featuring stoppages and knockouts but no official winner declared at the end.
But days before the fight at the Hard Rock Stadium, Paul claims Fury is interested in mixing it up with him and has sent him an intriguing proposal.
As per Paul, 'the Gypsy King' messaged him on Instagram pitching for a Pauls vs Fury double header that would see Tyson and Logan do battle, along with younger siblings Jake and Tommy - who have been involved in a spicy social media feud in recent months.
Asked about the one boxer he doesn't want to fight at a media scrum, Paul replied: "Tyson Fury. Tyson pitched me in my DMs. He's like, 'Yo, Jake could fight Tommy and you could fight me?!' I was like, 'Pass!'"
30-0 Fury is poised to face Deontay Wilder in a trilogy bout after an all-British unification title fight with Anthony Joshua fell through.
5-0 Tommy, meanwhile, is in action again this weekend but very much desperate to get in the ring with 'the Problem Child' in the near future.
"If Jake Paul gets in the ring with me it will be dangerous for him, but keep fighting these clowns he's fighting and it's not going to be dangerous for him," he said, as per DAZN.
"He is a cunning man. He has earned a lot of money and if there was any chance of Tyron Woodley beating Jake Paul that contract wouldn't have got signed.
"Why would Jake Paul put himself in a position where he is going to get knocked spark out? He is not going to fight anybody who will test him because he doesn't need to.
"He is making a fortune from YouTube. Why is he going to get his teeth knocked out for nothing?"
Promoter Frank Warren has reaffirmed his approval of the potential match-up, adding: "I will carry Tommy there for the fight. It's a no-brainer and the general public will watch it. Tommy also has a big following, is Tyson Fury's brother and brings a lot to the table."
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