  Given a fibration $f$ between two projective manifolds $X$ and $Y$, we
provide a sufficient condition such that the direct images
$f_{\ast}(K_{X/Y}\otimes L\otimes\mathscr{I}(f,\|L\|))$ is nef, where $L$ is a
holomorphic line bundle with non-negative relative Iitaka dimension and
$\mathscr{I}(f,\|L\|)$ is the relative asymptotic multiplier ideal sheaf.
