Speaking to Sky Sports afterwards, John Fury said: “They were disrespecting my son, the best ever heavyweight to wear a pair of boxing gloves.
“Coming out with all that rubbish, in my face, trying to be clever.
“So they stepped closer and stepped closer and at the end of it I am a warrior, that is what we do.
“If you come in the space, you are going to get what is coming.”
Son Tyson, meanwhile, said he did not see the incident and is only focused on Saturday’s fight.
“I was in the room doing interviews, but I’m not here for all that, I’m here to get the job done and go home and rest,” said the 35-year-old Briton.
Saudi authorities say no charges have been filed over the incident but Usyk’s manager Alexander Krassyuk says John Fury should apologise.
“It would be nice if we hear some apologies from John, because this was his behaviour,” Krassyuk told Sky.
“We are the example for the whole world, a new generation of kids are taking us as an example. What will they see from this?”