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There has been plenty of talk of a unification battle between WBC heavyweight champion Fury and Usyk – holder of the WBA-Super, IBF, WBO and IBO belts – since the latter beat Anthony Joshua for a second time in August.
Having made a U-turn on his decision to retire earlier this year, Fury beat Derek Chisora in a trilogy fight this month.
Plans for the 'Gypsy King' to meet Usyk next appeared to have been jeopardised when the Ukrainian was ordered to defend his WBA belt against mandatory challenger Daniel Dubois.
Frank Warren – who represents both Fury and Dubois – has insisted the huge unification bout will take place first and Arum claims there should not be long to wait for that fight.
"The two fighters have agreed to fight each other next," Arum told Sky Sports.
"With Fury and Usyk we're dealing with two adults, not a lot of back and forth. Usyk is a good friend of mine, he's very intelligent and Tyson is Superman, both as an athlete and as an intellect.
"So they want the fight. Both of them want the fight and so there'll be very little, if any, messing around. We'll be able to make that happen. I'm very, very confident.
"As I said, the fighters have both agreed to fight each other next without any interim fights. We'll have it all sorted out, I hope maybe by the end of the year."
Arum remains uncertain over the likely venue for the fight, though he described the prospect of being held at Wembley as "wonderful".
"Now the question is what's the date, and what's the site?" Arum said. "But that fight is definitely going to happen and it will happen in the first four months of next year.
"We are balancing a couple of significant offers from the Middle East, and also there's the possibility of doing the fight in the UK at Wembley with a massive 95,000 crowd in attendance.
"Fighters have a relatively short life and money is important, so if the money which has been proposed to us is real, that has to be taken into consideration.
"To go back to Wembley and do a fight before 95,000 people for me really stirs up the blood. It would be crazy. It would just be wonderful."