Joshua, who faces Wladimir Klitschko at Wembley later this month in front of a post-war record crowd for a fight in Britain of 90,000, insisted Fury's inactivity is the only reason they are yet to meet.
The current IBF heavyweight champion also insulted the Mancunian's "fat ass" as he responded to Fury labelling him a "poor man's Frank Bruno".
Comparing the muscular physiques of two fighters considered by some to be one-dimensional, the 28-year-old Fury wrote on Twitter: "Anthony Joshua is a poor man's Frank Bruno."
The 27-year-old responded by saying: "Typical Tyson Fury, always talking BS around fight time. Same offer as last time. Two ringside tickets, don't bring a plus one, it's for your fat ass."
Fury, who in November 2015 became the first fighter in 11 years to defeat Klitschko and transform the heavyweight picture into the one in which Joshua has thrived, then wrote: "I might be fat, but you're a paper champ and paper-chinned weightlifting rudeboy, wannabe badman.
"Know your place chump! Or I'll come back and (KO) you."
Finally, Joshua said: "Shut up and give me a date when you'll be back!
"You're the only reason we haven't had a fight yet! Be about what you say. I ain't going nowhere."
Terms for a rematch between Fury and Klitschko were twice agreed before the former was forced to postpone, first owing to an ankle injury and then to focus on recovering from depression.
Whether Fury will ever fight again remains in doubt, but in his absence Joshua has become one of the world's leading heavyweights.
It was earlier revealed on Tuesday, meanwhile, that a short film produced by Lucozade will be shown just before Joshua and Klitschko are to fight.
In it Joshua, who at Wembley also hopes to win the WBA title once held by Fury, revisits the journey that took him from teenage tearaway to Olympic and world heavyweight champion.
"I'm not usually one to look back but I liked the concept for the short film," he said. "It has been a real journey from growing up in Watford to fighting at Wembley Stadium.
"Everything in between; all the things I have done and all the people that have supported me, have made me who I am and this film illustrates that.
"I am excited to share my journey with my fans and I hope that it inspires the hunger in others."
