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The duo are set to square off at Madison Square Garden on August 6 in a huge grudge match.
However, Fury was denied entry to the US last week for a pre-fight press conference. The 23-year-old was turned away from Heathrow Airport in London due to an issue with his ESTA (Electronic System for Travel Authority).
Fury claims to have 'no idea' what he has done wrong, but Paul believes he is trying to 'weasel' out of the contest.
The YouTuber tweeted late on Sunday night: "Tommy is officially out by Wednesday morning if he doesn’t go to the embassy / come out of hiding. 3 other opponents lined up. I’m built different."
Paul did not name the potential opponents but he is hellbent on fighting in August after an extended period away from the ring.
His last fight was in December, when he was supposed to take on Fury before the former Love Island star pulled out with injury.
'The Problem Child' would face UFC welterweight champion Tyron Woodley instead and he defeated him via knockout.
Promoter Frank Warren, who also represents Tommy's brother WBC heavyweight champion Tyson, is keen for his client to make an appointment with the US Embassy and work out the problem.
He told iFL TV: "He [Tommy] has got to get to the Embassy, but no appointment has been made yet.
"I can understand where they're coming from. They've got a show to run or whatever and if we can't get it sorted out then obviously they'll go to another opponent so we'll just keep working on it and trying to make it happen."
KSI has offered Paul and Fury a lifeline and claimed they can fight on the undercard of his boxing return.
The rapper will face Alex Wassabi on August 27 and tweeted a message to longtime rival Paul on Wednesday.
He tweeted: "Clearly your card is dead in the water at this point. So lemme [sic] give you a lifeline.
"If you TRULY want to fight Tommy, fight on my card as the co-main event.
"I’ve got everything set up for you. Best believe you owe me though. Ball is in your court. #ItsAWinWin @tommytntfury."
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