  In this paper, we are interested in studying two properties related to the
denseness of the operators which attain their numerical radius: the
Bishop-Phelps-Bollob\'as point and operator properties for numerical radius
(BPBpp-nu and BPBop-nu, respectively). We prove that every Banach space with
micro-transitive norm and second numerical index strictly positive satisfy the
BPBpp-nu and that, if the numerical index of $X$ is 1, only one-dimensional
spaces enjoy it. On the other hand, we show that the BPBop-nu is a very
restrictive property: under some general assumptions, it holds only for
one-dimensional spaces. We also consider two weaker properties, the local
versions of BPBpp-nu and BPBop-nu, where the $\eta$ which appears in their
definition does not depend just on $\epsilon > 0$ but also on a state $(x,
x^*)$ or on a numerical radius one operator $T$. We address the relation
between the local BPBpp-nu and the strong subdifferentiability of the norm of
the space $X$. We show that finite dimensional spaces and $c_0$ are examples of
Banach spaces satisfying the local BPBpp-nu, and we exhibit an example of a
Banach space with strongly subdifferentiable norm failing it. We finish the
paper by showing that finite dimensional spaces satisfy the local BPBop-nu and
that, if $X$ has strictly positive numerical index and has the approximation
property, this property is equivalent to finite dimensionality.
