How do I get mold out of my bedsheets?

I have noticed tiny black spots on isolated areas of my bedsheets and pillow cases. They appear to be mold. They start out as pin-sized black spots and grow larger in a diamond shape. I have tried washing in hot water and bleach and they disappear for about a week and then begin to return. They don’t appear to be on mattress or pillows -just on sheets and more on the side where I sleep. Where did this come from and how do I get rid of it? I would toss the stuff, but it belongs to an expensive bed set, so I really want to save it.

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