Fury and Paul were due to settle their feud at New York's Madison Square Garden on August 6.
However, Fury was forced to withdraw from the bout after being denied entry to the United States.
Paul has now confirmed he will face Rahman Jr, the son of former heavyweight world champion Hasim Rahman, instead.
The fight will be the co-main event of the evening along with Amanda Serrano's featherweight world title fight against Brenda Carabajal.
Rahman Jr, 31, is a sparring partner of Paul and holds a professional record of 12 wins with six knockouts and just one defeat.
He has now been drafted in a late replacement after Paul confirmed his fight with Fury had been called off for the second time.
The pair were originally due to face each other in December before Fury withdrew from the bout with a broken rib and chest infection.
✅ Bigger— Jake Paul (@jakepaul) July 7, 2022
✅ More experienced
✅ KO power
But it doesn’t matter. They don’t have my heart, don’t have my drive & definitely don’t have my team.
August 6 I’m showing the world that Jake Paul is a bad mother fucker.
Press conference & ticket on-sale Tuesday 1pm ET. pic.twitter.com/sYAvCr0Cxu
"Fury's received a termination notice. MVP did everything it could 2 help him & team. He wasn't interested & he literally went into hiding," Paul posted on Twitter.
"2nd time in a row he has pulled out. 2nd time in a row I'm going to step up & take on a new opponent on short notice. AUG 6. MSG."
Paul later released a further statement saying he would be willing to travel to the UK to face Fury in the future.
"I'm confident you are a scared little boy who doesn't want to fight or understand business. I will pay you $500K to fight me in the UK," wrote Paul, who holds an unbeaten professional record of 5-0.
I wouldn’t get out of bed to fight you for $500k 🤣 Let me know when you’re ready to have a serious conversation. https://t.co/vYfDTqVsxv— Tommy Fury (@tommytntfury) July 7, 2022
"I will do it shortly after August 6. You accept?"
Fury turned down the offer, replying: "I wouldn't get out of bed to fight you for $500k.
"Let me know when you’re ready to have a serious conversation."
Paul has received praise on social media for accepting a fight with Rahman Jr, who he described as “bigger, stronger and more experienced” than Fury.
"Jake Paul has officially earned my respect, putting some of these so called 'boxers' to shame," wrote one fan.
Another added: "Okay respect to Jake Paul, not an easy fight."
However, others were more sceptical, with one Twitter user commenting: "[Rahman Jr’s] combined opponent record was 56-119 and he a sparring partner of Jake Paul?
"Lol here we go again."
Featured Image Credit: Jake Paul
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