Hearn, who promotes Joshua, said he had been informed of Miller's positive test by the Voluntary Anti-Doping Association (VADA), which works with athletes in boxing and mixed martial arts.
"We have been informed by VADA that there has been an adverse finding in Jarrell Miller's sample collected on March 20th, 2019," Hearn said on his Twitter account.
"We are working with all relevant parties and will update with more details soon. AJ's preparation continues for June 1st."
The likelihood of the fight taking place was further thrown into doubt later on Wednesday when the undefeated Miller was denied a boxing license by state boxing authorities.
"The New York State Athletic Commission has denied Jarrell Miller's professional boxing license application for a violation relating to the use of a prohibited substance," the commission said in a statement to ESPN on Wednesday.
"We have no further comment at this time."
Miller's camp have requested a B sample be tested, the BBC reported, and if that sample comes back negative Miller can apply for a licence again.
ESPN reported that Miller had tested positive for the banned substance GW1516, which boosts endurance and helps athletes burn fat, citing three sources familiar with the matter.
Miller's co-promoter Dmitriy Salita said the 30-year-old boxer would continue to train while his team sought more information.
"We are in the process of obtaining further information about VADA's finding and will have more to say soon on this developing situation," Salita told ESPN.
"In the meantime, Jarrell continues to train for his June 1 fight against Anthony Joshua."
VADA did not immediately respond to a Reuters e-mail requesting comment.
In February, Miller accused Joshua of using Therapeutic Use Exemptions (TUEs) to take testosterone that helped him build muscle, a charge that Joshua's camp denied.
(Reporting by Rohith Nair in Bengaluru and Rory Carroll in Los Angeles; Editing by Greg Stutchbury)
