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Fury and Wilder are set to meet for a third time later this year after 'The Gypsy King' dethroned 'The Bronze Bomber' of the WBC heavyweight title in February.
The American triggered the contracted rematch clause with a July date scheduled before the coronavirus pandemic.
A December showdown has been mooted but talks over a third bout has stalled with Fury eager to run it back so he can fight the likes of Whyte and unified heavyweight king Anthony Joshua.
On Twitter, the undefeated Brit wrote: "Deontay Wilder where you at mush? The clock is ticking!! I'm not gonna wait for ever!!! There is other bums in the division that I want to eat for breakfast."
.@bronzebomber where you at mush? The clock is ticking!! I'm not gonna wait for ever!!! There is other bums in the division that I want to eat for breakfast :egg: pic.twitter.com/ZZwhItBC2S
- TYSON FURY (@Tyson_Fury) August 20, 2020
Fury could be in line to defend his belt against Whyte who faces Alexander Povetkin on Saturday.
If victorious, 'The Body Snatcher' will become mandatory challenger for the WBC heavyweight championship.
However, Fury's British promoter Frank Warren insists his fighter won't be facing Whyte.
He said: "Tyson ain't fighting Dillian Whyte. That ain't going to happen this year and next year it ain't going to happen because he only wants to fight Anthony Joshua.
"Let's all be honest, if you were down the pub and you asked people 'what do you want, would you rather see Tyson fight Dillian Whyte or do you want to see him fight AJ?' It's a no-brainer, so let's get that on.
"Tyson Fury is not beholden onto Whyte. Tyson Fury is probably the biggest star in boxing at the moment and he will make his choices.
"Tyson against Joshua has to happen and I don't want it getting ruined by any other crap from Dillian.
"Of course Dillian wants his fight and he will get it, but that is like a pimple on a rhinoceros' a**e compared to seeing this fight."
Fury also has a commitment to Joshua after the pair of world champions agreed to a two-fight deal in 2021.