  This paper studies the approximation of singular Hermitian metrics on vector
bundles using smooth Hermitian metrics with Nakano semi-positive curvature on
Zariski open sets. We show that singular Hermitian metrics capable of this
approximation satisfy Nakano semi-positivity as defined through the
$\overline{\partial} $-equation with optimal $L^2$-estimates. Furthermore, for
a projective fibration $f \colon X \to Y$ with a line bundle $L$ on $X$, we
provide a specific condition under which the Narasimhan-Simha metric on the
direct image sheaf $f_{*}\mathcal{O}_{X}(K_{X/Y}+L)$ admits this approximation.
As an application, we establish several vanishing theorems.
