Hearn said the sample from Miller, scheduled to fight Joshua at Madison Square Gardens in New York on June 1, was collected on March 20 of this year.
Hearn tweeted: "We have been informed by VADA that there has been an adverse finding in Jarrell Miller's sample collected on March 20th, 2019.
"We are working with all relevant parties and will update with more details soon. AJ's preparation continues for June 1st at MSG."
Joshua and Miller were due to fight for the WBA, IBF, WBO and IBO titles in New York.
Although Joshua's preparations will continue the VADA news leaves the Miller fight in doubt and a replacement fighter may need to be found for what would be Joshua's USA debut.
Joshua retained his WBA, IBF, WBO and IBO titles when he stopped Alexander Povetkin in September.
Miller was kickboxing in 2014 when he tested positive for methylhexaneamine in a urine sample and was suspended for nine months by the California State Athletic Commission.
During their promotional tour for the fight, American Miller revealed that both fighters had agreed to between 10 and 12 weeks of drug testing in the build-up to the bout.
Joshua, during the London leg of the tour, said: "I am on (drug testing scheme) ADAMS whereabouts, they have to know where I am for an hour every day and could walk in anytime.
"They can turn up any time during the day and that is a lifetime commitment. I have been on that since 2012 and I pay £40,000-a-fight to be tested by VADA two times a week."