Structural modifications of single-walled aluminosilicate clay nanotubes have been studied under hydrostatic pressure by in situ synchrotron X-ray scattering. Imogolite nanotubes (INT) of nominal composition (OH)$_3$Al$_2$O$_3$SiOH, and methyl-modified imogolite nanotubes (m-INT), (OH)$_3$Al$_2$O$_3$SiCH$_3$, have hydrophilic and hydrophobic internal cavities, respectively. Nanotube chiralities also differ, with zigzag (INT) and armchair (m-INT) chirality. In this work, pressure-induced changes in nanotube morphology and atomic structure are studied as a function of chirality, affinity of the inner cavity, and the pressure-transmitting medium used. Radial deformation and collapse of nanotubes are evidenced below 3 GPa, followed by the formation of a lamellar phase at higher pressures. In the case of INT, the collapse pressure value depends on the pressure transmitting medium chosen. Axial compressibility is measured, and a pseudo Young's modulus Y is determined to be equal to ∼265 GPa for INT and below 80 GPa for m-INT, underpinning the role of nanotube chirality in mechanical properties.