On the frame bundle \({\mathcal F}(M)\) of an almost contact metric manifold (M,\(\phi\),\(\xi\),\(\eta\),g), we define an almost complex structure J and obtain that (\({\mathcal F}(M),g^ D,J)\) is an almost Hermitian manifold, where \(g^ D\) is the Sasaki-Mok metric induced on \({\mathcal F}(M)\). The integrability of the almost complex structure J and its relationship with the normality of the almost contact structure on M is studied. Moreover, we prove that (\({\mathcal F}(M),g^ D,J)\) cannot be either an almost Kähler or a nearly Kähler manifold unless it is a Kähler manifold.