Context. X-ray observations unveiled the existence of enigmatic point-like sources at the centre of young supernova remnants (SNRs). These sources, dubbed central compact objects (CCOs), are thought to be neutron stars formed by the supernova explosion. However, their multi-wavelength phenomenology is surprisingly different from that of most young neutron stars. Aims. The aim of this work is to understand the nature of the CCO 1WGA J1713.4-3949 in the G347.3-0.5 SNR through deep optical and infrared (IR) observations, the first ever performed for this source. Methods. By exploiting its derived Chandra X-ray position we carried out optical (BVI) observations with the New Technology Telescope (NTT) and adaptive optics IR (JHKs) observations with the Very Large Telescope (VLT). Results. We detected two faint ($I\approx23.5$, $I\approx 24.3$.) patchy objects close to the Chandra error circle in the NTT images. They were clearly resolved in our VLT  images which unveiled a total of six candidate counterparts ($17.8<H<20.3$) with quite red colours ($H-K_{\rm s}$ ~ 0.6). If they are stars, none of them can be associated with 1WGA J1713.4-3949 for the most likely values of distance and hydrogen column density. The identification of the faintest candidate with the neutron star itself can not be firmly excluded, while the identification with a fallback disk is ruled out by its non-detection in the I band. No other candidates are detected down to $B\sim 26$, $V\sim26.2$, $I\sim 24.7$, $H\sim$ 21.3, and $K\sim$ 20.5. Conclusions. Our high-resolution IR imaging unveiled a few objects close/within the Chandra X-ray position of 1WGA J1713.4-3949. However, at present none of them can be firmly identified as the object's likely counterpart.