ABSTRACT Since their discovery in 1991, carbon nanotubes (CNTs) – a novel one-dimensional carbon allotrope – have attracted considerable interest worldwide because of their potential technological applications such as electric and optical devices. In the astrophysical context, CNTs may be present in the interstellar space since many of the other allotropes of carbon (e.g. amorphous carbon, fullerenes, nanodiamonds, graphite, polycyclic aromatic hydrocarbons, and possibly graphene as well) are known to be widespread in the Universe, as revealed by pre-solar grains in carbonaceous primitive meteorites and/or by their fingerprint spectral features in astronomical spectra. In addition, there are also experimental and theoretical pathways to the formation of CNTs in the interstellar medium (ISM). In this work, we examine their possible presence in the ISM by comparing the observed interstellar extinction curve with the ultraviolet/optical absorption spectra experimentally obtained for single-walled CNTs of a wide range of diameters and chiralities. Based on the absence in the interstellar extinction curve of the ${\sim}$4.5 and 5.25 ${\rm eV} \ \pi$-plasmon absorption bands that are pronounced in the experimental spectra of CNTs, we place an upper limit of ${\sim}10\, {\rm ppm}$ of C/H (i.e. ${\sim}$4 per cent of the total interstellar C) on the interstellar CNT abundance.