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The two heavyweights are expected to do battle at some point in 2023 in a huge unification bout and came face to face after Fury defeated Derek Chisora earlier this month.
Speaking on his YouTube channel, Usyk called Fury by his middle name – a reference to a 2020 claim by Dillian Whyte, who said Fury had switched his first and middle names.
"When [the] cameras are off he's a different person," Usyk said.
"When cameras are on he's Tyson Fury, when they're off he's Luke Fury.
"Luke, I'm coming for you. I've been coming for him for a long time. I wish him a lot of health, happiness, love, patience and warm wind."
The unification bout is still fraught with complications, with the WBA ordering Usyk to defend his WBA Super belt against its regular champion Daniel Dubois.
But with Dubois and Fury both represented by Frank Warren that issue could still be resolved.
The anticipation of Fury-Usyk built after the former's win over Chisora when the latter came to the ring apron to get in his opponent's face.
"We negotiated it. If he agreed to fight, I would step onto the ring," Usyk said of the showdown.
"A lot of things were said since then, like he would slap me in my face. But apparently he changed his mind after having his buttocks burned."