  In this paper, we characterize the so called property $\textbf{L}_{o,o}$ as
defined by Dantas and Rueda Zoca, for compact, weak-weak continuous bilinear
maps. Motivated by this we weaken this property by defining the weak
$\textbf{L}_{o,o}$ for bilinear maps. We provide equivalence of the weak
$\textbf{L}_{o,o}$ property of $(X\hat{\otimes}_{\pi}Y,\mathbb{R})$ for linear
functionals and that of $(X,Y;\mathbb{R})$ for bilinear forms under certain
conditions on $X$ and $Y$. Moreover, we have also established that under
certain preassigned conditions, if a bilnear map $T$ belongs to a class which
satisfies the property $\textbf{L}_{o,o}$ (resp. the weak $\textbf{L}_{o,o}$)
for bilinear maps, then $T^*$ is a member of a class of operators which satisfy
the property $\textbf{L}_{o,o}$ (resp. the weak $\textbf{L}_{o,o}$) for
operators.
