  We prove that for a given Banach space $X$, the subset of norm attaining
Lipschitz functionals in $\mathrm{Lip}_0(X)$ is weakly dense but not strongly
dense. Then we introduce a weaker concept of directional norm attainment and
demonstrate that for a uniformly convex $X$ the set of directionally norm
attaining Lipschitz functionals is strongly dense in $\mathrm{Lip}_0(X)$ and,
moreover, that an analogue of the Bishop-Phelps-Bollob\'as theorem is valid.
