Yes. For the proof, see e.g. http://arxiv.org/abs/math/0609617
Theorem 4.1. Here a stronger result is actually proven:

Theorem: Let $(M, \omega)$ be a compact Kahler manifold,
and $Z\subset M$ a closed complex submanifold.
Denote by $[\omega]\in H^2(M)$ the Kahler class of $M$.
Consider a Kahler form $\omega_0$ on $Z$ such that
its Kahler class coinsides with the restriction
$[\omega]| Z$. Then there exists a
Kahler form $\omega'$ on $M$ in the same Kahler
class as $\omega$, such that $\omega|
Z=\omega_0$.

An additional cohomological assumption is needed, because we build a global extension, and for an extension to a local neighbourhood you don't need it; the positivity (needed in assumption) is achieved by adding a big multiple of a Kaehler form.