Haye is expected to return to the ring in December following his lengthy lay-off from the Achilles injury he suffered during his defeat to Bellew.
The 36-year-old said Joshua was not on his radar right now but vowed to beat the world heavyweight champion if given the opportunity.
‘AJ is not on my radar right now, but once I prove myself in December and am firing on all cylinders, I want the Joshua fight,’ Haye told Sky Sports.
‘If I get in the ring with him, I will find a way to beat that giant, he is 10 years younger, three inches taller and three stone bigger and I love and relish that kind of challenge.’
Responding to Haye’s bold claim, Bellew told Express Sport: ‘Do I need David Haye? No. Does David Haye need me? Yes. Without a shadow of a doubt.
‘I’ve seen an article this week saying he’s talking about Anthony Joshua. He must be on the drugs.
‘This guy Joshua is a colossal monster and David has come out saying “me and him could be the biggest fight ever”.
‘He’s insane. He really is.’
Bellew has not fought since his impressive victory over Haye at the O2 but was recently called out by British UFC middleweight champion Michael Bisping.
‘If I fought Michael Bisping in a boxing ring, I would spark him clean out in about 40 seconds,’ Bellew, 34, said.
‘I am not Floyd Mayweather and he is not Conor McGregor. I wouldn’t carry a man for 10 rounds. I would absolutely smash him.
‘It would be so one-sided that people would laugh. So the only way to make this a competitive fight should be to do it in the Octagon.
‘I’d maybe have a roll around with him for a couple of minutes and then I’d probably still choke him out.’