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Published 14:14 26 Jun 2016 BST
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If found guilty, Fury could be stripped of his world titles and face a lengthy suspension from the sport.
The Mirror also reports that Fury's younger cousin Hughie tested positive for the same substance.
Both fighters have denied any wrongdoing in a short statement released on Sunday.
The statement reads: "We are baffled by today's story in the Sunday Mirror. Tyson Fury denies any allegation of doping. He looks forward to recovering from his injury and defending his titles against Wladimir Klitschko in October."
The samples are said to have been collected last February or March and Fury was made aware of the presence of the banned substance in September.
It's understood that both Fury and his cousin are to be called before an independent panel where they will answer the case.
Fury was initially expected to meet former opponent Klitschko again on July 9 in Manchester but pulled out of the bout with an injury.

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