Winslow, who leads the Jets with 17 receptions this season, will be required to sit out the team's next four games, Newsday reported.
He said the news he tested positive for a banned substance "came as a huge surprise to me."
"I have investigated all possible sources since learning of the positive test, including possible medical causes, but frustratingly, I do not know the source," he said in a statement. "I do not take supplements that list any banned ingredients, but the NFL policy is very strict, so I am accountable for the results."
"Guys, we got to look at it this way: We have a really good tight end in Jeff Cumberland," Ryan said of Winslow's absence. "The guy's a young stud tight end."
Winslow will next be eligible to play Nov. 17 in Buffalo.