Fury beat his old British rival Derek Chisora to retain his WBC belt earlier this month and immediately called out Ukrainian Usyk, who holds the IBF, IBO, WBO and WBA titles and was ringside at the Tottenham Hotspur Stadium.
Later Fury said the bout will probably take place in Saudi Arabia in February or March, although the date and venue has yet to be announced.
"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 [rubbish] 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.
"Now the question is what's the date and what's the site. 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 Mid East and also there's the possibility of doing the fight in the UK at Wembley with a massive 95,000 crowd in attendance."