After taking the latest step on his road to the undisputed heavyweight title, Joshua told the WBC champion he was ''down to ride whenever, wherever'' and would ''knock him spark out''.
Joshua claimed all he needed to do to beat Wilder would be to ''get him in the ring, I'll knock him spark out''.
He said: "Are you asking me if I want to become undisputed champion of the world? IBO, IBF, WBA, WBO – 21 professional fights, six world titles fights. Does that not show how good I'm going? Forget the hype, I'm about business, let's get the business done."
He added: "As Dillian [Whyte] said [last week]: 'What's up Wilder, let's go baby, let's go!'"
Fury had provided commentary for the fight at the Principality Stadium on Sunday on his Instagram story but the videos seem to have since been removed.
He said: "Just watching these two and I'm telling you they're a pair of wooden tops.
"They're not even good enough to be my sparring partners, the pair of them.
"They're a pair of bums. No panache, no flair, no show. Waste of money.
"There's only one Tyson Fury for a reason. This is Mac year 2018, they're a pack of bums.
"Parker is what Parker is - he's a young lad trying.
"AJ's a wooden top - he ain't no superstar, I am the man."
After giving his thoughts on round 10, Fury added: "I could beat AJ in my comeback, treble a jab up, swing the right hand up, downstairs, upstairs, ain't good enough to be a sparring partner."
He concluded: "That was a s**t fight, a s**t performance by both fighters.
"Listen AJ, please don't fight Wilder because he'll knock you spark out and take my £50million. Let me break you and your little bum."
Warning: This video contains offensive language
