  Fast X-ray transients (FXTs) are X-ray flares lasting minutes to hours.
Multi-wavelength counterparts to these FXTs have been proven hard to find. As a
result distance measurements are through indirect methods such as host galaxy
identification. The three main models proposed for FXTs; supernova shock
breakout emission (SN SBO), binary neutron star (BNS) mergers and tidal
dirsuption events (TDEs) of an intermediate mass black hole (IMBH) disrupting a
white dwarf (WD), have a different associated L$_{X, peak}$. Therefore
obtaining the distance to FXTs will be a powerful probe to investigate the
nature of these FXTs. We use VLT/MUSE observations of a sample of FXTs to
report the redshift of between 13 and 22 galaxies per FXT and use these
redshifts to calculate the distance, L$_{X, peak}$ and the projected offsets.
We find L$_{X, peak}>10^{44}$ erg s$^{-1}$ if we assume any of the sources with
a redshift measurement is the true host galaxy of the corresponding FXT. For
XRT 100831 we find a very faint galaxy within the 1$\sigma$ uncertainty region
with a chance alignment probability of 0.04. For XRT 060207 we find a candidate
host galaxy at z = 0.939 with a low chance alignment probability. However, we
also report the detection of a late-type star within the 3$\sigma$ uncertainty
region with a similar chance alignment probability. For the remaining FXTs we
find no sources within their 3$\sigma$ uncertainty regions. We rule out a SN
SBO nature for all FXTs based on L$_{X, peak}$ and the projected offsets. For
XRT 100831 we conclude the detected galaxy within the 1$\sigma$ uncertainty
position is likely to be the host galaxy of this FXT. From the available
information, we are not able to determine if XRT 060207 originated from the
galaxy found within 1$\sigma$ of the FXT position or was due to a flare from
the late-type star detected within the 3$\sigma$ uncertainty region.
