  Let $(V,\omega)$ be a compact K\"ahler manifold such that $V$ admits a cover
by Zariski-open Stein sets with the property that $\omega$ has a strictly
plurisubharmonic exhaustive potential on each element of the cover. If
$X\subset V$ is an analytic subvariety, we prove that any
$\omega|_X$-plurisubharmonic function on $X$ extends to a
$\omega$-plurisubharmonic function on $V$.
  This result generalizes a previous result of ours on the extension of
singular metrics of ample line bundles. It allows one to show that any
transcendental K\"ahler class in the real Neron-Severi space $NS_{\mathbb
R}(V)$ has this extension property.
