The bout was originally for February 17th in Riyadh but was postponed on Friday after Fury suffered a cut during sparring.
The "Ring of Fire" bout was set to unify Fury's WBC heavyweight championship with Usyk's WBA, IBF, WBO and IBO belts, crowning the first unified heavyweight boxing champion since 1999.
Fury won his WBC championship after knocking out American Deontay Wilder in 2020, while Usyk won his belts after outscoring Britain's Anthony Joshua in 2021.
Both boxers are undefeated in their professional careers.
Saudi entertainment head and organiser Turki Alalshikh told a special Saturday edition of the MMA Hour with Ariel Helwani on YouTube that he was confident the fight would happen on the new date.
"This fight I want it to happen," he said. "On 18th of May we will have Tyson-Usyk... on the 18th we will have the answer."
Alalshikh said May 18th also met a deadline set by the IBF for the fight to happen.
Both Fury and Usyk appeared on the show via video links, Fury in Saudi Arabia and Usyk from his training camp in Spain, and said they had agreed on the new date.
Fury explained that he had arranged to spar 12 rounds with four different boxers and suffered the injury in round five, requiring a hospital trip and 11 stitches.
"They have the medical reports, they have everything... nothing much anybody can do about getting a cut in sparring. Shit happens. Never had one before. First time for everything," he said.
Fury also brushed aside a suggestion from Usyk's camp that he was looking for a way out.
"Why would I not want to fight for the biggest payday in my life?" he asked.
"I've put 10 weeks in a training camp. Why would I not want to fight?"
Usyk said his reaction to the news of the postponement was "just smile. I just smile and go to training."
Saudi organisers Riyadh Season said on Friday that tickets for the original fight would be refunded.
