  We show that the set of bounded linear operators from $X$ to $X$ admits a
Bishop-Phelps-Bollob\'as type theorem for numerical radius whenever $X$ is
$\ell_1(\mathbb{C})$ or $c_0(\mathbb{C})$. As an essential tool we provide two
constructive versions of the classical Bishop-Phelps-Bollob\'as theorem for
$\ell_1(\mathbb{C})$.
