Let R be a 2-torsion free $sigma$-prime ring with an involution $sigma$, U a nonzero square closed $sigma$-Lie ideal, Z(R) the center of Rand d a derivation of R. In this paper, it is proved that d = 0 or $U;{subseteq};Z(R)$ if one of the following conditions holds: (1) $d(xy);-;xy;{in};Z(R)$ or $d(xy);-;yx;{in}Z(R)$ for all x, $y;{in};U$. (2) $d(x);{circ};d(y);=;0$ or $d(x);{circ};d(y);=;x;{circ};y$ for all x, $y;{in};U$ and d commutes with $sigma$.