We review the existing distance estimates to the black hole X-ray binary Swift J1727.8$-$1613 with a discussion of the accuracies and caveats of the associated methodologies. As part of this, we present new line-of-sight HI absorption spectra captured using the MeerKAT radio telescope. We estimate a maximum radial velocity with respect to the local standard of rest of $24.8 \pm 2.8$ km s$^{-1}$, which is significantly lower than that found towards an extragalactic reference source. Given the location of Swift J1727.8$-$1613 at Galactic longitude and latitude $(l, b) \approx (8.6\deg, 10.3\deg)$, we explore the feasibility of the HI absorption method. From this we derive a near kinematic distance of $d_{\rm{near}} = 3.6 \pm 0.3\ (stat) \pm 2.3\ (sys)$ kpc as lower bound for the distance to Swift J1727.8$-$1613. We compare our results with those derived from different distance determination methods including the use of colour excess or reddening along the line of sight, which we constrain to $E(B-V) = 0.37 \pm 0.01\ (stat) \pm 0.025\ (sys)$ using near-UV spectra. By combining this with donor star magnitudes reported by Mata S\'anchez et al. (2024b), we suggest an increased distance of $5.5^{+1.4}_{-1.1}$ kpc, which would imply a natal kick velocity of $190 \pm 30$ km s$^{-1}$.
Comment: 12 pages, 4 figures, 2 tables, submitted to ApJL