  Fast X-ray Transients (FXTs) are X-ray flares with a duration 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, and a binary neutron star merger. We present the X-ray light
curve and spectrum, and 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 May 6, 2021, thirteen days after the onset of the flare.
No transient optical counterpart is found in the 1" (3$\sigma$) X-ray
uncertainty region of the source to a depth $g_{s}$=27.0 AB mag. A candidate
host lies within the 1" X-ray uncertainty region with a magnitude of 25.9 $\pm$
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 $z_{\rm spec}=1.5082 \pm 0.0001$
and an offset of 4.2$\pm$1" (37$\pm$9 kpc) from the FXT and another one with
$z_{\rm phot}=1.04^{+0.22}_{-0.14}$, at an offset of 3.6$\pm$1" (30$\pm$8 kpc).
Based on the properties of all the prospective hosts we favour a binary neutron
star merger, as previously suggested in the literature, as explanation for XRT
210423.
