  We study the Bishop-Phelps-Bollob\'as property for numerical radius
restricted to the case of compact operators (BPBp-nu for compact operators in
short). We show that $C_0(L)$ spaces have the BPBp-nu for compact operators for
every Hausdorff topological locally compact space $L$. To this end, on the one
hand, we provide some techniques allowing to pass the BPBp-nu for compact
operators from subspaces to the whole space and, on the other hand, we prove
some strong approximation property of $C_0(L)$ spaces and their duals. Besides,
we also show that real Hilbert spaces and isometric preduals of $\ell_1$ have
the BPBp-nu for compact operators.
