  We introduce a weakened notion of norm attainment for bounded linear
operators between Banach spaces which we call \emph{quasi norm attaining
operators}. An operator $T\colon X \longrightarrow Y$ between the Banach spaces
$X$ and $Y$ is quasi norm attaining if there is a sequence $(x_n)$ of norm one
elements in $X$ such that $(Tx_n)$ converges to some $u\in Y$ with
$\|u\|=\|T\|$. Norm attaining operators in the usual sense (i.e.\ operators for
which there is a point in the unit ball where the norm of its image equals the
norm of the operator) and compact operators satisfy this definition. The main
result of the paper is that strong Radon-Nikod\'ym operators such as weakly
compact operators can be approximated by quasi norm attaining operators (even
by a stronger version of the definition), which does not hold for norm
attaining operators. This allows us to give characterizations of the
Radon-Nikod\'ym property in term of the denseness of quasi norm attaining
operators for both domain spaces and range spaces, extending previous results
by Bourgain and Huff. We also present positive and negative results on the
denseness of quasi norm attaining operators, characterize both finite
dimensionality and reflexivity in terms of quasi norm attaining operators,
discuss conditions to obtain that quasi norm attaining operators are actually
norm attaining, study the relationship with the norm attainment of the adjoint
operator, and present some stability properties. We finish the paper with some
open questions.
