  In this paper, we introduce the notion Lie-derivation. This concept
generalizes derivations for non-Lie Leibniz algebras. We study these
Lie-derivations in the case where their image is contained in the Lie-center,
call them Lie-central derivations. We provide a characterization of Lie-stem
Leibniz algebras by their Lie-central derivations, and prove several properties
of the Lie algebra of Lie-central derivations for Lie-nilpotent Leibniz
algebras of class 2. We also introduce ${\sf ID}_*-Lie$-derivations. A ${\sf
ID}_*-Lie$-derivation of a Leibniz algebra G is a Lie-derivation of G in which
the image is contained in the second term of the lower Lie-central series of G,
and that vanishes on Lie-central elements. We provide an upperbound for the
dimension of the Lie algebra $ID_*^{Lie}(G)$ of $ID_*Lie$-derivation of G, and
prove that the sets $ID_*^{Lie}(G)$ and $ID_*^{Lie}(G)$ are isomorphic for any
two Lie-isoclinic Leibniz algebras G and Q.
