Inelastic nuclear interaction probability of 400 GeV/c protons interacting with bent silicon crystals was investigated, in particular for both types of crystals installed at the CERN Large Hadron Collider for beam collimation purposes. In comparison to amorphous scattering interaction, in planar channeling this probability is ∼36% for the quasi-mosaic type (planes (111)), and ∼27% for the strip type (planes (110)). Moreover, the absolute inelastic nuclear interaction probability in the axial channeling orientation, along the ⟨110⟩ axis, was estimated for the first time, finding a value of 0.6% for a crystal 2 mm long along the beam direction, with a bending angle of 55 μrad. This value is more than two times lower with respect to the planar channeling orientation of the same crystal, and increases with the vertical angular misalignment. Finally, the correlation between the inelastic nuclear interaction probability in the planar channeling and the silicon crystal curvature is reported.