  Very recently, motivated by the result of Bhatia and \v{S}emrl which
characterizes the Birkhoff-James orthogonality of operators on a finite
dimensional Hilbert space in terms of norm attaining points, the
Bhatia-\v{S}emrl property was introduced. The main purpose of this article is
to study the denseness of the set of multilinear maps with the Bhatia-\v{S}emrl
property which is contained in the set of norm attaining ones. Contrary to the
most of previous results which were shown for operators on real Banach spaces,
we prove the denseness for multilinear maps on some complex Banach spaces. We
also show that the denseness of operators does not hold when the domain space
is $c_0$ for arbitrary range. Moreover, we find plenty of Banach spaces $Y$
such that only the zero operator has the Bhatia-\v{S}emrl property in the space
of operators from $c_0$ to $Y$.
