The Bishop-Phelps-Bollob\'as property for numerical radius in ${\mathrm{\ell }}_1(\mathbb{C})$

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 ${\mathrm{\ell }}_1(\mathbb{C})$ or $c_0(\mathbb{C})$. As an essential tool we provide two constructive versions of the classical Bishop-Phelps-Bollob'as theorem for ${\mathrm{\ell }}_1(\mathbb{C})$.