  We study the denseness of Crawford number attaining operators on Banach
spaces. Mainly, we prove that if a Banach space has the RNP, then the set of
Crawford number attaining operators is dense in the space of bounded linear
operators. We also see among others that the set of Crawford number attaining
operators may be dense in the space of all bounded linear operators while they
do not coincide, by observing the case of compact operators when the Banach
space has a 1-unconditional basis. Furthermore, we show a
Bishop-Phelps-Bollob\'as type property for the Crawford number for certain
Banach spaces, and we finally discuss some difficulties and possible problems
on the topic.
