It is known that the only Einstein real hypersurface in a quaternionic projective space \(QP^m\), \(m \geq 2\), is a geodesic hypersphere of a certain radius. Moreover, there are no Einstein real hypersurfaces in quaternionic hyperbolic space \(QH^m\), \(m \geq 2\). In this paper the authors impose a condition on the Ricci tensor that is slightly weaker than the Einstein condition. More precisely, let \(M\) be a real hypersurface of \(QP^m\) or \(QH^m\), and denote by \(D\) the maximal quaternionic subbundle of the tangent bundle of \(M\). Then \(M\) is said to be \(D\)-Einstein if its Ricci tensor restricted to \(D\) is a multiple of the identity at each point. The authors derive a complete classification of \(D\)-Einstein real hypersurfaces in \(QP^m\) and \(QH^m\).