Bellew, who recently defeated David Haye via an eleventh round stoppage, posted a picture of himself standing next to former world ruler Fury from some years back, prompting a response from his latest rival.
Fury questioned whether the fans would like to see the fight, stirring up interest in what looks to be another mammoth task for Bellew, who was widely written off before his shock win over Haye.
“I didn’t say anything. I just put a picture up – no one is being bright – this clown last week labelled me as a politician. I was honoured,” Bellew told talkSPORT.
“It was exactly what I put with the emoji flicking his chin – thinking – it’s just a thought, a thought for now.
“You’ve asked me would I fight Tyson? Why wouldn’t I?”
As retirement remains a big possibility on the back of a career-best paycheck, Bellew is adamant any decision will be his and nobody could have any arguments to the contrary should he walk away with his head held high.
“I don’t feel a duty to do anything besides getting the kids from school. I don’t have [any] obligations to this sport – I don’t owe anyone anything – I don’t owe anyone nothing.
“The only people I owe something to (are) those kids because I put them on this planet – they didn’t choose to be here – I put them here. Believe it or not, they’re the only thing I’m obligated to.”
And on his potential to win the sport’s biggest prize, Bellew, who is a rematch target for Haye, added: “There’s a lot of things being said about me and let’s just remember on Friday night just gone, David Haye was the most avoided heavyweight in the world, he was the bogeyman of the heavyweights – no one dared mention his name.
“Only two people wanted to fight David Haye and I was one of them, Shannon Briggs is the other one. We’ve got something in common the pair of us – we’ve both got screws loose.
“No champions even whispered David’s name. Yet I go in and do him, and now every single champion wants to know who Tony Bellew is. Because you know why? Because he’s just a fat cruiserweight champion.
“But you know what? Underestimate me again at your peril. I’m not sure what the next move is. But would I like to be heavyweight champion of the world? Yeah right, I would.
“Imagine that: the fat guy winning the heavyweight championship as well as the cruiserweight. Is that realistic? Is it too dangerous? I don’t know – time will tell.”