Fast X-ray Transients (FXTs) are X-ray flares with durations ranging from a few hundred seconds to a few hours. Possible origins include the tidal disruption of a white dwarf by an intermediate-mass black hole, a supernova shock breakout, or a binary neutron star merger. We present the X-ray light curve and spectrum as well as deep optical imaging of the FXT XRT 210423, which has been suggested to be powered by a magnetar produced in a binary neutron star merger. Our Very Large Telescope and Gran Telescopio Canarias (GTC) observations began on 2021 May 6, thirteen days after the onset of the flare. No transient optical counterpart is found in the 1.″ (3σ) X-ray uncertainty region of the source to a depth g s = 27.0 AB mag. (We use the word "counterpart" for any transient light in a wave band other than the original X-ray detection wave band, whereas the word "host" refers to the host galaxy.) A candidate host lies within the 1.″ X-ray uncertainty region with a magnitude of 25.9 ± 0.1 in the GTC/HiPERCAM g s filter. Due to its faintness, it was not detected in other bands, precluding a photometric redshift determination. We detect two additional candidate host galaxies: one with z spec = 1.5082 ± 0.0001 and an offset of 4.″2 ± 1.″ (37 ± 9 kpc) from the FXT, and another one with ${z}_{\mathrm{phot}}={1.04}_{-0.14}^{+0.22}$ and an offset of 3.″6 ± 1.″ (30 ± 8 kpc). Based on the properties of all the prospective hosts, we favor a binary neutron star merger, as previously suggested in the literature, as the explanation for XRT 210423.