Probing for the host galaxies of the fast X-ray transients XRT 000519 and XRT 110103

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 $R>25.8$.