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Fury won the titles against Wladimir Klitschko in 2015, however, he subsequently relinquished them amid failed drugs tests and struggles with depression.
Appearing on This Morning, the newly-crowned WBC heavyweight champion described Joshua's titles as his 'leftovers'.
He said: "I've already been the undisputed heavyweight champion of the world. When I beat Klitschko I had all the belts, so as far as I'm concerned Anthony Joshua just has my leftovers.
"I never lost those belts, I had to take time out to deal with mental health issues, so they're my belts. Until a man is defeated in a boxing ring, how can you pretend to be the champion?"
Fighting talk from The Gypsy King then.
Has anyone seen @Tyson_Fury?
Here's what Wednesday's #ThisMorning looks like - from :keycap_ten:!@Schofe @hollywills pic.twitter.com/nWaDbEht4b
- This Morning (@thismorning) March 4, 2020
No doubt a unification clash between the two is what the people want to see, and Fury suggested that may be his last fight after his rematch with Deontay Wilder.
He said: "I've got two more fights left, then we're going to really think about what we do from there.
"How long does a piece of string go on? I'm undefeated in 31 professional fights and this is my 12th year as a professional."
Wilder triggered his rematch clause over the weekend, following his seventh round defeat to Fury last month, but the 31-year-old doesn't appear overly worried about that.
He said: "I beat him the first time and I beat him the second time and I will surely beat him the third time.
"He's a very worthy opponent. He's a very dangerous opponent and he has the eraser knockout.
"You're only one mistake away from disaster, you have to be switched on for all 12 rounds."
Fury also revealed that he is in the process of setting up a boxing club to 'encourage kids to follow their dreams'.
He said: "I'm opening a local boxing club now where I am, I'm in the process of doing it.
"Like I had a chance to walking into a boxing club as a kid and do what I did today, I want to give other kids that opportunity.
"I just want to be positive and encourage kids to follow their dreams like I followed mine.
"So many people said it was impossible to do, so I know what it's like to be told you can't do something.
"I want to tell people everything is possible if you put your mind to it."
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