For terahertz (THz) wave applications, tunable and rapid modulation is highly required. When studied by means of optical pump-terahertz probe spectroscopy, single-walled carbon nanotubes (SWCNTs) thin films demonstrated ultrafast carrier recombination lifetimes with a high relative change in the signal under optical excitation, making them promising candidates for high-speed modulators. Here, combination of SWCNTthin films and stretchable substrates facilitated studies of the SWCNT mechanical properties under strain,and enabled the development of a new type of an opto-mechanical modulator. By applying a certain strain to the SWCNT films, the effective sheet conductance and therefore modulation depth can be fine-tuned to optimize the designed modulator. Modulators exhibited a photoconductivity change of 3-4 orders of magnitude under the strain due to the structural modification in the SWCNT network. Stretching was used to control the THz signal with a modulation depth of around 100 % without strain and 65 % at a high strainoperation of 40 %. The sensitivity of modulators to beam polarisation is also shown, which might also come in handy for the design of a stretchable polariser. Our results give a fundamental grounding for the design of high-sensitivity stretchable devices based on SWCNT films.