  For Banach spaces $X$ and $Y$, a bounded linear operator $T\colon X
\longrightarrow Y^*$ is said to weak-star quasi attain its norm if the
$\sigma(Y^*,Y)$-closure of the image by $T$ of the unit ball of $X$ intersects
the sphere of radius $\|T\|$ centred at the origin in $Y^*$. This notion is
inspired by the quasi-norm attainment of operators introduced and studied in
\cite{CCJM}. As a main result, we prove that the set of weak-star quasi norm
attaining operators is dense in the space of bounded linear operators
regardless of the choice of the Banach spaces, furthermore, that the
approximating operator can be chosen with additional properties. This allows us
to distinguish the properties of weak-star quasi norm attaining operators from
those of quasi norm attaining operators. It is also shown that, under certain
conditions, weak-star quasi norm attaining operators share numbers of
equivalent properties with other types of norm attaining operators, but that
there are also a number of situations in which they behave differently from the
others.
