Buddhism denies atman, a permanent unchanging self. Some Buddha-nature texts describe what is within as permanent, blissful, self and pure, using precisely the four terms the earlier tradition had rejected. The Mahaparinirvana Sutra does this deliberately. The tension is not an outsider's invention; it is internal, and Buddhist thinkers have worked on it for most of two millennia.
The main lines of resolution are three. Buddha-nature is potential rather than an existing entity, so nothing permanent is asserted. Buddha-nature is emptiness stated positively, the mind's lack of inherent defilement described in affirmative language for the encouragement of practitioners; this is the standard Tibetan Gelug reading, and it treats the affirmative texts as provisional in meaning rather than definitive. Or Buddha-nature is the mind's luminous nature, a claim about the character of awareness rather than about a possessed thing, which some trace to an early canonical passage describing the mind as luminous and defiled only by adventitious stains.
Tibetan Buddhism divides sharply here. The shentong or emptiness-of-other view, associated with the Jonang school and influential in parts of Kagyu and Nyingma, holds that the awakened nature is empty of everything extrinsic but not empty of its own qualities. The rangtong view holds that it is empty in the same way as everything else. The Jonang position was suppressed politically in seventeenth-century Tibet, which indicates the stakes.
The comparative temptation is to read Buddha-nature as a soul, or as the divine spark of Christian and Platonic mysticism, or as atman in Advaita. Buddhist commentators from Indian scholiasts to modern teachers have resisted all three, and a reader who wants to understand the doctrine on its own terms should note that the resistance is itself a large part of the tradition's literature on the subject.