  There are two notions of approximate Birkhoff-James orthogonality in a normed
space. We characterize both the notions of approximate Birkhoff-James
orthogonality in the space of bounded linear operators defined on a normed
space. A complete characterization of approximate Birkhoff-James orthogonality
in the space of bounded linear operators defined on Hilbert space of any
dimension is obtained which improves on the recent result by Chmieli\'nski et
al. [ J. Chmieli\'nski, T. Stypula and P. W\'ojcik, \textit{Approximate
orthogonality in normed spaces and its applications}, Linear Algebra and its
Applications, \textbf{531} (2017), 305--317.], in which they characterized
approximate Birkhoff-James orthogonality of linear operators on finite
dimensional Hilbert space and also of compact operators on any Hilbert space.
