Whyte appeared to be in line to face one of the leading names in the decision after defeating Joseph Parker, but Fury and Wilder are both now otherwise engaged after agreeing to face each other, most likely in Las Vegas in either November or December.
The Jamaican-born fighter has repeatedly taken to his social media pages in recent days to criticise the duo, however, prompting a response from Fury when he posted a video on Instagram on Sunday of a mock kiss between the former unified world champion and WBC title holder Wilder.
"Embarrassing bunch of cowards," Whyte wrote. "Let's go chumps #clowns"
A lengthy Fury reply mocked Whyte's recent results, saying: "Let's face facts here. You struggled with Dereck Chisora and you should of (sic) lost that by at least four rounds, then Joseph Parker nearly knocked you out in round 12."
Whyte came back with a counter, uploading a second post on Instagram, while also repeating the same words on Twitter.
"Unlike you, I'll fight anyone, @Tyson_Fury," he wrote. "You refused to fight me as I knocked you down so many times in sparring.
"If you pull out in November with yet another fake injury, I'm more than happy to take your place and smash up that other coward @BronzeBomber [Wilder]."
Unlike you I’ll fight anyone. @Tyson_Fury u refused to fight me as I knocked you down so many times in sparring and if you pull out in November with yet another fake injury I’m more than happy to take your place and smash up that other coward @BronzeBomber
— Dillian Whyte (@DillianWhyte) August 20, 2018
Photo Getty Images. Caption: Dillian Whyte celebrates victory