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The 32-year-old defensive end told ESPN he had failed the test after unwittingly consuming a banned substance in an over-the-counter supplement.
The National Football League has yet to formally issue a statement on the case.
"Few things are more important to me than my name and reputation. This might call that into question for some, which has me heartbroken," Ninkovich told ESPN.
"I don't want to cut any corners. I want to do things the right way, with high integrity, and that's what I have always wanted to stand for."
Ninkovich has played 116 consecutive games for the Patriots since joining the team in 2009. He was due to miss the start of this campaign after suffering a torn triceps muscle in a practice session.
Last year Ninkovich made 52 tackles, 6.5 sacks and seven pass deflections during the regular season.
"Any supplement I've ever used was bought at a store. I was unaware something I bought had a substance in it that would give me a positive test because it wasn't listed (as an ingredient)," Ninkovich told ESPN.
"One thing I have learned is that if a supplement is not NSF certified, there are no regulations that ensure that what is on the label is 100 percent accurate. That is a hard lesson for me to learn at this stage in my career, but I take responsibility for it. It's a mistake I made and it hurts that I won't be there for my teammates."