Using the same warp method, are the upper left corner coordinates different or the same with bilinear and NN? Are the output pixel sizes the same? If they are the same, I would think that the georeferencing (the warp grid) is the same and the effect you see is from the resampling method since one is an average and one using the original value. This can make boundary DN values appear to be 'shifted' when they are not.
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