"Anthony Joshua is a poor man's Frank Bruno", wrote the Morecambe-based heavyweight.
Joshua promptly hit back at Fury, who was stripped of his IBF world heavyweight title in 2016 after refusing to fight mandatory challenger Vyacheslav Glazkov.
"Typical Tyson Fury always talking BS around fight time. Same offer as last time. x2 ringside tickets, don't bring a plus one - it's for your fat a**," Joshua wrote on Twitter.
Fury is currently in training as he prepares to make his long-awaited comeback after a serious of drug and fitness issues.
He replied: "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 OK you."
Joshua then appeared to lose patience with his heavyweight rival: "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 u say. I ain't going nowhere."
Fury then hit back for one final time: "You shouldn't throw stones when you live in a glass house, deal with b****ko first then I'll give you a boxing lesson, I promise. Good luck chump."
