It emerged on Tuesday that the WBA had ordered Joshua to agree a date for a mandatory title defence against Alexander Povetkin.
Eddie Hearn, Joshua's promoter, questioned whether Wilder wants to step into the ring with the Briton, stating that he had not heard back from the American's camp after sending a contract over last week.
Hearn said it appears that a bout with Povetkin will be next for WBO, IBF, IBO and WBA strap-holder Joshua and Wilder – the WBC champion – let his feelings be known over that latest twist.
He tweeted: "You coward a** b****. Had the world waiting for 3 months playing games just for this moment. You're not a true champion [‼️] you're just a weak minded coward that's holding hard metal. #Facts."
Wilder continued: "All they had to do is say they were scared to fight period. But instead made themselves look like what they are, not stand up guys.
"If your [sic] scared say your [sic] scared. He can just be the champ of England.
"I'm embarrassed for them. He's a true coward plain and simple."