Fury, the half-brother of world heavyweight champion Tyson Fury, is set to face YouTube celebrity Paul on August 6.
The 23-year-old posted a video on Instagram on Tuesday, recorded on Monday evening, where he explained his application to enter the United States had been rejected.
“So I just want to come on here and set the record straight before anybody else tries to, me and my team this morning arrived at Heathrow airport ready for the press conference, ready to fly out,” he said.
“As soon as I entered the airport I got pulled to one side and I was told by a Homeland Security officer that was there, that my ESTA has been denied and I wasn’t able to travel to the USA for a reason that I apparently know.
“I can stand here and say I have done absolutely nothing wrong and I have no clue why I’m not allowed to travel to the USA.
“I’ve been training for a fight this whole time and that’s all I’ve been doing. I have no clue why they will not allow me to travel today and neither does any of my team or my lawyers.
“So now I’m having to go to the embassies and all this sort of stuff, trying to resolve it and I’m in the middle of training, guys, I don’t know why this has happened today – it is a massive shock to me and my whole team.”
The issue casts doubt on the rearranged bout, set for Madison Square Garden in August, after a rib injury had seen Fury pull out of their scheduled fight last December.
It was claimed earlier this month that Tyson Fury had also been denied access to the United States over alleged links to alleged crime boss Daniel Kinahan, although the WBC world heavyweight champion has distanced himself from those connections.
Paul responded to the news on Twitter, saying: “Tommy, no matter how hard you try to get out of this fight, I’m going to do everything in my power to not let you weasel your way out.
“My team and my partners have made it clear the steps you need to take to solve your latest excuse.”