  Let $(X,\omega)$ be a compact K\"{a}hler manifold with a K\"{a}hler form
$\omega$ of complex dimension $n$, and $V\subset X$ is a compact complex
submanifold of positive dimension $k<n$. Suppose that $V$ can be embedded in
$X$ as a zero section of a holomorphic vector bundle or rank $n-k$ over $V$.
Let $\varphi$ be a strictly $\omega|_V$-psh function on $V$. In this paper, we
prove that there is a strictly $\omega$-psh function $\Phi$ on $X$, such that
$\Phi|_V=\varphi$. This result gives a partial answer to an open problem raised
by Collins-Tosatti and Dinew-Guedj-Zeriahi, for the case of K\"{a}hler
currents. We also discuss possible extensions of K\"ahler currents in a big
class.
