Paul and Fury were on course to collide at Madison Square Garden on 6 August with the two scheduled to appear in New York two weeks ago for a press conference.
The Briton however has stated he was denied entry to the United States, leading to the fight falling through for the second time in seven months.
Paul announced on social media this week the Fury fight was officially off, sparking a search for a new opponent a month away from fight night.
Rahman, the son of the former heavyweight champion, is now set to step into the ring for what will be Paul’s sixth professional fight.
Unlike Paul’s previous opponents, Rahman Jr is an active professional boxer, having fought 13 times.
The 31-year-old has won 12 of those bouts, six of those coming via knockout, with his only loss coming in his last outing against James McKenzie Morrison in April.
While Paul’s focus is now on the American, he is extended new offer to Fury, offering the Briton $500,000 to take a fight in the UK.
Paul tweeted on Thursday: ‘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. I will do it shortly after 6 August. You accept?’
Fury had previously insisted he remains confident of the fight happening, offering to take it to a ‘neutral location’ to avoid any more problems.
MORE : Jake Paul terminates Tommy Fury fight contract and accuses rival of ‘hiding’
MORE : Jake Paul insists he is moving on from Tommy Fury fight and accuses Tyson Fury of backing out of $1million bet
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