  Over the past few years, $\sim$30 extragalactic fast X-ray transients (FXRTs)
have been discovered, mainly in Chandra and XMM-Newton data. Their nature
remains unclear, with proposed origins including a double neutron star merger,
a tidal disruption event involving an intermediate-mass black hole and a white
dwarf, or a supernova shock breakout. A decisive differentiation between these
three promising mechanisms for their origin requires an understanding of the
FXRT energetics, environments, and/or host properties. We present optical
observations obtained with the Very Large Telescope for the FXRTs XRT 000519
and XRT 110103 and Gran Telescopio Canarias observations for XRT 000519
designed to search for host galaxies of these FXRTs. In the $g_s$, $r_s$ and
$R$-band images, we detect an extended source on the North-West side of the
$\sim$ $1^{\prime\prime}$ (68% confidence) error circle of the X-ray position
of XRT 000519 with a Kron magnitude of $g_s=$26.29$\pm$0.09 (AB magnitude). We
discuss the XRT 000519 association with the probable host candidate for various
possible distances, and we conclude that if XRT 000519 is associated with the
host candidate a supernova shock breakout scenario is likely excluded. No host
galaxy is found near XRT 110103 down to a limiting magnitude of $R>25.8$.
