Russell, a member of the West Indies squad for next week's World Twenty20 championship in India, could be banned for up to two years if found guilty, the Jamaica Gleaner reported.
The Jamaica Anti-Doping Commission (JADCO) declined to comment when contacted by AFP to confirm the report. Russell, 27, also could not be reached for comment.
Failing to be available for a test is regarded as the equivalent of testing positive for a performance-enhancing substance, meaning Russell faces a long spell on the sidelines if his violation is confirmed.
He has regularly played in the West Indies one-day squads and has also enjoyed stints with various T20 sides in India, Australia and Pakistan.
If Russell's doping sanction is confirmed, it would spell further disruption to the West Indies World T20 preparations, which has been dogged by a series of withdrawals and a bitter pay dispute.