Barrett said in an article in a May issue of Swimming World magazine that Hosszu's performances, which dwarfed those of all other swimmers, were "incredible".
Barrett, who swam at the 1996 Atlanta Olympics, said he had no "hard proof" of any misdemeanours however.
The article prompted an indignant response from the 26-year-old Hosszu who vehemently denied taking drugs and told a news conference in a Budapest hotel on Tuesday that her legal team was formulating action against Swimming World and Barrett.
"I never took performance-enhancing substances," she said. "I would not sell my soul for money, fame or victory.
"Cheating is completely alien to me as is shrugging off a self-serving attempt to undermine my credibility by someone completely unknown to me.
"I will tell Mr Barrett at the appropriate time and place that he made a grave mistake, he offended me grossly, and hurt the sport. His accusations are wholly unfounded."
Hosszu has won her signature medley event at every major meeting she has entered since the 2012 Olympics and been dominant on the professional circuit, becoming the first swimmer last year to earn more than $1 million in prize money.
In his article, Barrett said: "No one competes, consistently, at a higher level than she does. Repeat -- no one, ever.
"Not Michael Phelps or Katie Ledecky and certainly not Ryan Lochte," he added referring to the three American greats.
"Her consistency, her ability to recover, and her never-flagging form continues without breakdown."
Tamas Gyarfas, FINA vice-president and chairman of the Hungarian Swimming Association, said if results alone merited suspicion then questions would be asked of people like 18-times Olympic champion Phelps.
"We never questioned the superhuman results of Phelps, so why do they do this now?" added Gyarfas.
"I think what Phelps did was great. This is why there are tests -- and Katinka goes to the toilet (to give urine samples) more than old men suffering from a bad prostate."
(Reporting by Marton Dunai; Editing by Ian Chadband and Ken Ferris)
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