The BBC and newspapers said the positive test had come after the death of the 25-year-old England international's newborn child.
"There is no question it was a mitigating factor," Hull manager Steve Bruce said last month. "Jake has kept all of the problems that he had to himself. All footballers think they are macho men but they have problems like everybody else."
The BBC said a Football Association disciplinary hearing had taken place last week at which it was decided Livermore would not be banned.
There was no immediate comment from the club, relegated to the Championship (second tier) at the end of last season.
Image: Jake Livermore
Photograph: Tony Marshall/Getty Images
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