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Published 14:43 6 Jun 2016 BST
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"The two of us are speaking at the moment and we'll see where it goes but I want to fight the 'Haye-faker'," the 33-year-old said.
Bellew, whose victory over Makabu extended his career record to 27-2-1 with 17 knockouts, has been critical of the quality of Haye's opponents since the former two-weight world champion returned to the ring.
"I'd say Hayemaker but it's 'Haye-faker' because he's pulled out of that many fights over the last few years. He ran scared - absolutely petrified from Tyson Fury - not once, but twice," he added.
Fury has already ruled out a fight with Haye after those two withdrawals, attributed to injuries to Haye's head and foot respectively.
The 35-year-old is understood to be keen on a bout with Anthony Joshua, whose next IBF heavyweight title defence comes against Dominic Breazale on June 25. He told JOE a meeting in 2017 would be realistic.
The bout against Briggs will be Haye's third of the year after victories over Mark de Mori and Arnold Gjergjaj, and the Bermondsey-born fighter has previously suggested he could be in shape for a fourth fight before the end of 2016.
Haye admits he would be keen to cut his weight if a fight does come to fruition, having weighed in at 227lb for his comeback victory over De Mori and 224lb for the two-round demolition of Gjergjaj.

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