No. Bitmaps (.bmp) are defined by X/Y pixels and a page size; they do not contain a dpi description. The X/Y pixels and the page size are interpolated by applications such as Windows Explorer to determine the dpi. Various applications may perform different rounding techniques and therefore may show different dpi values. These differences are beyond the creating applications control.
For example: A bitmap exported from ArcMap at a resolution of 300 dpi sometimes has a resolution listed as 299 dpi in Windows Explorer.