National Basketball League chairman Sam Rossiter-Stead declined to comment, and referred questions to Drug Free Sport NZ.
Drug Free Sport NZ chief executive Graeme Steel said the organisation does not comment on cases unless a player is found guilty by the Sports Tribunal, under the process set by the World Anti-Doping Code.
When players returned positive tests, Drug Free Sport NZ would put a case before the tribunal, which would determine whether there had been a rule violation.
Sharks' general manager Jill Bolger did not immediately return calls.
Dawson, 25, played his first year for the Sharks in 2010, and returned last year, when he was part of the NBL championship winning team.
The drug scandal comes as three other Sharks' players are accused of assault after a punch-up outside a New Plymouth bar on Sunday morning.
Three players, aged 19, 21 and 28, have been charged and are to appear in the New Plymouth District Court on Wednesday.
The incident came after the team's win against the Taranaki Mountain Airs.
The positive test to Gareth Dawson's B sample was extremely disappointing to the Southland Sharks basketball team, Sharks' board spokesman Gareth Davis said tonight in a statement.
"Gareth's positive test is extremely disappointing to us. The drug in question was prescribed to him to treat a pre-existing condition and the Board and management were unaware of this prescription."
Mr Davis said Dawson had chosen to stand aside from the team while the B sample was tested, but is now suspended until the Drug Free Sport NZ hearing at a date to be confirmed.
"We had a duty of good faith as an employer because there was the medical issue, but now it is in the hands of Drug Free Sport NZ, Gareth has been stood down from the Sharks squad," Mr Davis said.
He said players in the squad had been reminded of the need to advise Sharks' management of any prescription medicines they were taking.
"We are also making sure they tell their doctor they may be drug tested as an athlete so that this issue can be avoided in future," he said.
Mr Davis confirmed Dawson was tested out of competition by Drug Free Sport NZ in February 2014 and returned a positive test to a banned substance.