Following a three city tour last week, Fury has not enrolled in the program, which is the drug testing arm of the WBC, and not signing up will end up costing him the title, even if he wins.
Wilder has since announced he will fight Tyson Fury in the United States on December 1, while Joshua's representatives have been pushing for Wilder to reach a conditional agreement to fight Joshua next year.
"If he does not enroll, then the WBC can not sanction the fight", Sulaiman exclusively told Sky Sports.
"Fury has promised me a part in the program".
The Briton, who like Wilder has an unbeaten record, said after the victory that he would prefer to take on the American next, but was happy to fight whoever came his way.
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With a little less than two months until the fight, a testing program tailored for the Wilder-Fury fight would start right around the same time that these programs normally would.
Wilder knocked out Luis Ortiz in the 10th round on March 3, but only after that fight had originally been scheduled to take place in November 2017 was canceled - and later rescheduled - when Ortiz tested positive for two banned diuretics, chlorothaizide and hydrochlorothiazide, which he claimed he used to treat high blood pressure.
Wilder (40-0, 39 KOs), who will be making his eighth title defense, has always been a strong advocate for a testing program specific to each of his fights, in part because he has seen three previous bouts canceled or had to find a new opponent because his original foe failed a drug test.
Fury (27-0, 19 KOs) has a history with VADA.
Joshua's promoter Eddie Hearn and his father Barry want the agreement in place before Wilder faces Fury.