We propose to obtain relativistic near-single-cycle optical vortices carrying orbital angular momentum through the post-compression of Laguerre-Gaussian pulses in gas-filled multipass cells. Our simulations revealed that 30 fs optical vortex pulses centered around 800 nm with a pulse energy of millijoule level can be compressed to near-single-cycle duration with topological charges from 1 to 20 within an argon-filled cell with five passes. The spectral broadening preserves the topological charge of the input beam; the spatio-spectral couplings are also discussed. The energy of the vortex pulses could be scaled up by increasing the dimensions of the cell. The relativistic near-single-cycle vortices are of great interest for the generation of ultrashort helical electron bunches based on hybrid electron acceleration in underdense plasmas and on isolated relativistic extreme ultraviolet optical vortices from high-order harmonic generation in solid foils. (C) 2020 Optical Society of America